Intuitionist mathematics claims that mathematics is purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles existing in an objective reality. [0]
In intuitionist mathematics there is only potential infinity, no actual infinity. Constructive set theory differs from Zermelo set theory.
That has many consequences in practice. Applying intuitionist mathematics to physics we can come to the conclusion that time flows and it helps reconcile quantum mechanics with general relativity.[1]
> Applying intuitionist mathematics to physics we can come to the conclusion that time flows
Can you expound on this a little more? I'm completely new to the idea of intuitionistic mathematics and much more so to its applications in physics; the constructive approach to thinking about objects and properties is very refreshing and I'd like to hear how you've related those principles with the paradox of time in the context of classical physics.
Unfortunately no. I got the info from Quanta Magazine article. I am not knowledgeable enough to respond on this. Even though I did some high level math courses in University, I wasn't interested enough.
I am sure other HN users understand this stuff better.
From the wikipedia source:
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.
And Im wodering why Im not looking more into the philosophy of math, i find this interesting because I am mainly an intuition driven person. Any ideas for courses online?
The mathematical definition of intuition is not the colloquial definition. For example, in intuitionistic mathematics, you can't proof by contradiction, which means that some things that are "intuitively true" in the colloquial sense, are inaccessible.
The reason why intuitionist mathematics is useful in QM is because it can be viewed as resource based logic.
But remember that (as far as I know) you can do intuitionist mathematics in classical mathematics, but not the other way around. So you can think of intuitionist mathematics as being embedded in classical mathematics.
I think in your first paragraph you're confusing intuitionistic logic and linear logic --- intuitionistic logic still lets you "clone" and "delete" propositions freely.
The second paragraph is somewhat true --- at its base, the modern use of the term "intuitionistic logic" refers merely to not accepting the law of the excluded middle. However, there are varieties of intuitionism that are very, very inconsistent with classical logic, like those adopting the idea that "every function between real numbers is continuous".
When you separate internal and external logic, for example in topos theory, the external logic is still classical.
For example, any topos has internal logic on subobjects of at least a Heyting algebra. In some cases you specialise to a Boolean algebra.
However, when you write things down such as your arguments around functors and constructions, you argue according to classical mathematics externally. When you enter into the category (which is chosen not to be Boolean) and argue on the subobject structure, then you are entering intuitionistic logic. This is what I mean when I say that classical logic can be argued to have intuitionistic logic embedded in it.
The cardinality of the reals being equal to the cardinality of the naturals is then something you have to construct inside the topos. So you need to construct along the spirit of a natural numbers object (perhaps a real numbers object) and then use the internal logic to argue about continuity.
I think you are right about the comment about linear logic. I must be confusing A U A with A U ^A.
I checked the paper that I was thinking about and the condition that I saw there is distributivity being dropped rather than double negation being dropped.
I think you're both right in a certain sense. The set of provable statements in intuitionist logic is a subset of what is provable in classical logic, which is what I believe the parent was referring to.
You're certainly right that any constructively provable statement is classically provable, that follows from intuitionism/constructivism generalizing classical logic. But I don't see how we could interpret the following to mean that.
> you can do intuitionist mathematics in classical mathematics
> But remember that (as far as I know) you can do intuitionist mathematics in classical mathematics, but not the other way around.
There are a number of embeddings of classical logic into intuitionistic logic. The most popular/obvious is the double-negation embedding that maps a proposition to its double negation.
Why do you think the same thing doesn't apply to theories? Since you mention topos theory, the double-negation topology is actually a basic construction in topos theory, used to construct a dense Boolean subtopos of a given topos.
You can always generate substructures with closure properties. The subobject structure comes from all the subobjects—so it is the default logic of the topos. Of course you can do the same thing here as you mentioned since it is a Heyting algebra at the least.
My point is something else: I don't know of a way to have a intuitionistic exterior logic and a classical internal logic.
I've always thought math as we know it is a result of both deep underlying relationships between natural constructs and how we perceive them. For example, a species who has no sense of vision will not develop geometry the same way we would. A species that has a sense of vision that also includes a direct distance perception (as opposed to our stereoscopic vision) will probably come up with a very different form of geometry.
Though I'd like to think that most species would come up with some version of calculus, even though the notation will be obviously different. Afterall, two of our own species did so independently.
Combining the words "most" and "species" feels a little bit silly. Among all the millions of species on this planet only humans have displayed the intelligence necessary to contemplate any of this.
That's only true if you limit this to conscious thought. Lots of organisms perform impressively complex calculations. The desert ant odometer is one classic example:
Yes, but it's possible to perceive distict objects without vision, except the separation is in time, rather than space. Or, depends on the number of sense organs available (e.g. a blind person can sense two different objects simultaneously by holding one in each hand).
Agreed, I wonder if the same amount of abstract concept would be possible though, can you feel a graph? could you do trigonometry with just hearing and sense of touch?
"could you do trigonometry with just hearing and sense of touch?"
Can you get data on this from a Monte Carlo method of study on match
play between sighted and blind players playing go and chess at distance
and blind which requires a mental model of the board state and
translations?
"two of our own species [discovered calculus]"..? Which two species are those? I assume by "own own" you mean Terran species, and one of them is Homo Sapiens, which is the other one? Really curious...
I'm not convinced that blind mathematicians count as their own species, nor that they echolocate, and so it doesn't seem to say much about his theory one way or the other.
We could perhaps modify the theory, thinking of the mechanics of research. Progress is made by an individual in the niches where others have not managed yet. Therefore the advantage in geometry for a blind mathematician may be in a different mental model of spaces.
It is not surprising at all that almost all blind mathematicians are geometers. The spatial intuition that sighted people have is based on the image of the world that is projected on their retinas; thus it is a two (and not three) dimensional image that is analysed in the brain of a sighted person. A blind person’s spatial intuition on the other hand, is primarily the result of tile and operational experience. It is also deeper – in the literal as well as the metaphorical sense. […]
recent biomathematical studies have shown that the deepest mathematical structures, such as topological structures, are innate, whereas finer structures, such as linear structures are acquired. Thus, at first, the blind person who regains his sight does not distinguish a square from a circle: He only sees their topological equivalence. In contrast, he immediately sees that a torus is not a sphere […]
Maybe vision is helpful with elementary geometry. I don't know how vision would help with differential geometry, geometry of affine spaces, topology, algebraic geometry.
Maybe visualization is more helpful, but you can visualize things without seeing.
A sense of depth and distance, or any kind of metric is helpful, and senses like vision and hearing might be helpful in forming that, but I'm not sure they are required.
From another point of view, math theories are congruent, and you can explain the same things without using geometry, but using algebra, or differential equations or number theory and so on. You certainly can explain any math concept using just modern set theory - maybe it is not the most convenient thing to do.
A species that has a sense of vision that also includes a direct distance perception (as opposed to our stereoscopic vision) will probably come up with a very different form of geometry.
Except once axioms have been fixed, the resulting geometry will be the same wether you
perceive reality via visible light or echolocation.
I really liked this perspective from Bartosz Milewski in this (very good) podcast episode: mathematics is not inherent to the world, it is inherent to the human brain (paraphrased, around minute 32 in the episode)
And really one of the biggest pieces of proof of this is that some of the earliest mathematics developed was planar geometry.
But, planar geometry (primarily involving triangles and squares) does not actually exist in nature. Which means that planar geometry is an approximation technique developed by the brain in order to begin to understand the actual much more complex geometries that appear in the real world.
This also suggests that mathematics is 100% constructed by the human brain, even if it is highly-influenced by relationships found in the physical world.
But, this makes sense because we really don't ask the same question about human language. We almost never ask: is human language constructed or discovered?
>This also suggests that mathematics is 100% constructed by the human brain, even if it is highly-influenced by relationships found in the physical world.
As I see it, mathematics is both discovered and invented.
We can model every existing thing in every possible world using math. Even if both the set of all things that might exist and all possible mathematical constructions are infinite, the later is larger. That's because we can also construct mathematical models of things that doesn't exist.
So it looks that from the set of all possible mathematical constructions, we extracted a subset that maps to objects in reality. That looks like a discovery process.
But we also constructed mathematical models of things that don't have corespondents in reality, so that much be more of an invention process.
Let's pretend for a bit that we forget all what we know and tomorrow we will start inventing things again. Or that a species of aliens start fresh on a planet.
The mathematic theories and notions we and the aliens might discover, build or invent might be different than the theories and notions we know today but would probably be equivalent. That suggests that mathematics just exists somewhere in its own world waiting to be discovered.
I too, subscribe to this view. In my mind, reality is nothing more than a vast integer variable field with interactions between some variables. All our mental models, from vision to math are attempts at approximating that complexity down to a level that our brain's limited processing capacity can handle...
Why integers one might ask. For me, even rational numbers are such an approximation.
This. It is discovered in the same sense that humans "discover" facts about humans via science- we're not born with perfect self-knowledge.
It is "created" because there is no guarantee that our efforts correspond to reality. In that sense we are just playing in the sandbox of what we can conceive.
The fact that mathematics corresponds so "unreasonably" to objective reality is because what we call objective reality is mediated by our brain's own idiosyncrasies and limitations of thinking.
It's no more surprising that external objective reality is mathematically predictable and describable than it is that our eyes can see shapes and some, but not all, light. That's the purpose of eyes and they tell us enough of what we need to know that we can survive.
Our brains are exactly the same thing- they tell us a story in a way which helps us to survive. Actual reality may be beyond our capacity to conceive of or worse, may seem like nonsense to us because it's aggressively illogical or specifically contradictory and reality therefore makes no "sense" to us.
>It is "created" because there is no guarantee that our efforts correspond to reality. In that sense we are just playing in the sandbox of what we can conceive.
But doesn't this dance kinda near the notion that I am a brain in a vat and none of you exist (read this question with me as the speaker or with yourself as the speaker)?
I would say our brains do have limits. I can't well envision a 4D object. But once we reduce things down to simple logical axioms and constructs, these exist as much as anything can be said to exist. Even if I was a simulation that didn't even have a brain, much less eyes, the concepts I come up with would exist more than the flesh I incorrectly thought I had.
>>But once we reduce things down to simple logical axioms and constructs, these exist as much as anything can be said to exist.
I agree.
But this assumes logic and at least the principle of non-contradiction: not both A and not A - is how reality "really" is.
We can't get past the idea that it must be this way because only provable nonsense lies on the other side of this assumption. But our brains may be fundamentally unable to process ultimate reality and the nonsense a contradiction represents may be a statement not about reality but our brains, our thinking.
So logical contradictions aren't actually nonsense, they're the sound of us hitting the walls of what our minds can conceive of. All animal have such limits. We assume those limits look like darkness- stuff we can't peer into. What if they look like impossibility instead?
That's the point of view I'm entertaining here. I am not saying this is true. It certainly isn't useful or provable as far as I know, but it is possible.
The practical value of such an exercise, if it has any (and I think it does) is to twofold.
One seems to expand my imagination to the maximum extent pops me out of the assumptions that frame my thinking and this seeps into my thinking about things, technical problems, generally.
Two it confers humility and a certain openess and makes me less judgmental. So that, for example, when I hear or read people with claims to spiritual knowledge I don't automatically blow them off as crazy / bitter / ignorant because what they're saying "makes no sense".
Thinking and talking about Ultimate Reality capital U capital R, ought to fill us all with humility if we're being intellectually honest by our own standards. Yet, I find people totally lack that humility. They make huge pronouncements about Ultimate Reality which they can't really be sure of, and the effect this has on the world, and how we think of each other, and therefore how we treat each other, and even the effect on one's own mind, is one of diminishment generally.
Mea culpa, I was one of those people and I didn't like it.
If you get down to the core of knowledge and how we know something, this is what's really there, and it's good to be reminded of it.
Not exactly the same but 2-dimensional beings on a curved surface can discover the 3rd dimension using math, we did this proof in a differential geometry class.
Ctrl-F for "2D" (2-dimensional creatures), although I can't find the "discovery", maybe I remember it wrong.
Btw, this was one of the mindfucks I learned studying physics. Other ones are:
- dimensional analysis: checking if formulas can be wrong, and more importantly, finding formulas (eg, period of a pendulum) just by analyzing units that can be involved (no constants obv)
- similarly, rate of growths in dimensions: why giants can't exist (bone resistance is proportional to section area and load (weight) is proportional to volume
- Conservation laws derive easily from symmetries (energy from invariance in time, momentum from invariance in direction etc)
I think you may be talking about the curvature of the surface (the Riemann tensor). It is intrinsic, i.e. does not depend on the notion of the surface being embedded into a higher-dimensional space.
Assuming a species is intelligent enough to discover math, if they have some significantly different way of experiencing the world I think they would take a different path but would eventually reach the same destination we are heading towards, likely merging with us by the point we are at now.
I could see the layman understanding of geometry being quite different than ours, limited by their own visual system. But their mathematicians and our mathematicians would discover the same math, through different paths and perhaps with different strengths and weaknesses and a different path of educating budding mathematicians from the grade school level to the post doc level.
Tell me more about the difference between "potential infinity" and "actual infinity".
Recursive functions in computer programming, or fractals in maths, are infinite but can be described. Each small part looks like the whole, but the whole can never be fully known within a finite universe. Is that "potential" or "actual" infinity?
I've been thinking a lot about infinity this past week, mostly because of Conway's lectures. I'm trying to figure out whether the following logic makes sense.
- Assuming that the universe is finite
-> Infinity cannot be approximated with finite numbers (MIP* = RE) [1]
-> Numbers cannot have infinite strings of digits
-> The future can never be perfectly preordained (Gisin) [2]
-> Free will exists (Conway) [3]
This is not a proof that free will exists! But I think this is a consistency proof that free will is finite.
Now you're telling me that there's another kind of infinity, and I want to know what that means for this thought.
His definition of free will is quite limited. He only suggests that free will is limited to a "free" choice between options within one's power.
Conway said that there either is determinism or free will. If the future can be perfectly preordained (Gisin) due to infinite possibilities, then there is no free will. But if there are finite options, I think that there is "free will" for particles and people to choose between those options.
Free will is a philosophical concept which is essentially dualist. It's an attempt to reconcile two subjective experiences - the feeling of making a decision, and the fact that sometimes we can't predict the actions of others - with universal determinism.
There's no reason why universal determinism and free will need to be related. So trying to "prove" free will with math makes as much sense as "proving" free will with weather forecasting.
Even if that weren't true you still can't get to Conway's Step 3, because "free will" could just be statistical noise, and not willed in any sense at all.
>Free will is a philosophical concept which is essentially dualist.
Is it though? I came to the conclusion that one could consider free will the self-realization of a person in space-time -- in other words, free will is the only and only choice one can make (at each decision they face), that they make because of who they are thus far.
In that sense, "free will" is the same as the "person" (or the person's essense) -- and would it make sense for it to be any other way? Randomness wouldn't be free will, and having a second entity (e.g. a soul in dualism) do the decision making just moves the question and degree upwards (from "how the person has free will?" to "how the person's soul has free will?")
If a person faced with a choice A or B could go either way, then I wouldn't call that really free will (even though for some reason that's what most people have in mind when talking about the subject).
Either a person is a concrete personality/being and that manifests through its choices (which means that when faced with the A or B dillema they can only chose one or the other based on who they are), or they have some degree of randomness in their decision making (which makes them less of a person in my eyes -- randomness is not tied to our person, it is, as the name reveals, random).
When some people say "free will" they actually mean "free (independent) from the person". But that's not a will then -- a will denotes something being tied to a person (a person is their will).
You're conflating different kinds of randomness here.
A coin flip and a hand of cards are both random, but obviously cards are 'more random'.
This can be quantified - 'more random' is information complexity, or information entropy. (Two names for the same thing.)
If free will exists, then it must necessarily be a singularity of infinite information complexity, which is the same thing as a singularity of infinite randomness.
> Free will is a philosophical concept which is essentially dualist.
It's not, although many people think this. Consider that the majority of philosophers are actually Compatibilists, in which free will is compatible with determinism, and so they would disagree with your definition. If the majority of practicing experts disagrees with your view on their subject, it's probably time to revise your view.
First, I would take strong issue at your describing the majority of philosophers as Compatiblists. I can find no support for such a broad statement. Also, one of the primary criticisms of Compatiblism, dating back as early as Kant, is that what Compatibilists define as ‘free will’ is not free will as most people, including philosophers understand it. Pulling a quick quote, the Compatibilist position can typically expressed by the following:
Arthur Schopenhauer famously said, "Man can do what he wills but he cannot will what he wills." In other words, although an agent may often be free to act according to a motive, the nature of that motive is determined.
This position that the ‘free will’ people experience is the ability to ‘decide’ to take some and then take it, while then saying the the decision to do so was predetermined, is not what is meant (especially in the vernacular) by free will.
Therefore, while I will not support a metaphysical dualist basis for ‘free will’ it is clearly incorrect to dismiss an argument of such based on 1) an appeal to authority which is unsupportable (in that your claim about the position of a majority of ‘experts’ seems unsupported), and 2) that even if it was supported, that said appeal to authority is a valid refutation of an argument. Further, dismissal of an argument by choosing to substitute a contested definitional term (the meaning of free will) for one that is clearly not the same is not rhetorically valid.
> Arthur Schopenhauer famously said, "Man can do what he wills but he cannot will what he wills." In other words, although an agent may often be free to act according to a motive, the nature of that motive is determined.
You're assuming this is relevant. Turns out, it's not.
> Also, one of the primary criticisms of Compatiblism, dating back as early as Kant, is that what Compatibilists define as ‘free will’ is not free will as most people, including philosophers understand it.
Nobody definitively understands free will. Some people conjecture it has certain properties, mainly incompatibilists. So far they have mostly been wrong.
> This position that the ‘free will’ people experience is the ability to ‘decide’ to take some and then take it, while then saying the the decision to do so was predetermined, is not what is meant (especially in the vernacular) by free will.
Potential infinities are just saying, you don’t have an infinity in your hand — you just have a recipe for arbitrarily large numbers (or arbitrary accuracy on real numbers, etc).
Eg, the “natural numbers” are a recipe to make as many as you want (by taking the successor of the last one) but not an actual (as in existing) infinite collection — the only ones which exist are the ones you construct, and the “potential infinity” refers to the fact that you can always make a new one.
"Actual infinity" is merely an abstraction, not unlike many others. It is very useful. Abstract thinking is part of what separates humans from (other) animals.
While there is technically no proof, there is a lot of evidence that the universe is infinite. In particular it's curvature is flat within the margin of measurement error which implies an infinite universe outside the observable universe.
> Intuitionist mathematics claims that mathematics is purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles existing in an objective reality. [0]
The "mental activity" part isn't strictly necessary. A constructive mathematics will produce results that are largely the same. What matters most is that mathematical objects are proven by construction, and so proofs have a direct translation to computation.
I think you and OP are talking about different things because the term "intuitionist" has been used for both the philosophy that EJ Brouwer espoused (which matches what OP said) and the large field of constructive mathematics that has been greatly influenced by Brouwer's philosophy but has embraced approaches like formal logic which Brouwer himself disliked.
I was merely pointing out that the "mental" dependency of intuitionism isn't strictly necessary. Take it out, and you're left with an equally valid constructive logic.
On the question about what is the reality, what explanation is the true one, Merrifield said that the Math works on all of the explanations and what those explanations do is simply to model the behaviour of nature and not necessarily reflect the reality.
That's how Newtonian physics and relativistic physics are both correct models, tools to model nature and simply can be used to whenever suitable.
Wouldn't that mean that mathematics is just an invented tool to reason about physical models?
Well yes and no. A lot of mathematics found its roots in trying to model the real world, however I would then argue that the truths we prove about these models are truly discoveries. And with these discoveries we can often generalize the setting to more abstract formulations, independent of the physical reality.
I think that this can be seen in the fact that sometimes mathematicians and especially physicists can reason about objects that they are not sure about what the right definition should be. Many mathematicians reasoned about continuous functions long before we had a concrete definition of them. But when Cauchy introduced the definition and Weierstrass proved it is equivalent to preserving limits (which was the intuition at the time), we had not truly discovered something new and mathematical.
This whole idea was then generalized to topology when it was shown that pre images of continuous functions preserve the "openess" of sets, i.e. we realized that no concept of distance was not needed to describe continuity, which is very surprising.
The expression is invented but the underlying relationships are discovered.
It's silly to think that the sqaure root of negative one is something real to be discovered. It's just a stand-in to express complex relationships more succinctly. The same for negative numbers, for that matter.
It would be equally silly to think that the underlying relationships are invented. Nobody invented prime numbers, they were discovered.
Complex numbers aren't "necessary for the continuum" as you put it, and some realists might argue that they don't hold the same "discoverability" as the reals.
I wouldn't. I think the reals and the complex numbers have the same "realness" and that neither represents any innate property of the physical world, despite how obviously useful they are in physical models.
> If you throw away real numbers, then you lose major things in physics. I think for example you lose the wave function in QM.
Then again, have you ever observed a non-computable number as a component of the value of a wave function in nature? Real numbers add a lot of cruft which can never be practically observed (due to not being computable).
One of the main reasons we lose things is that our results have been built upon the real numbers since they were easier to conceptualize and to work with, but it's possible that a lot can be recovered (maybe even everything we care about) using only the computable numbers. For instance, see https://en.wikipedia.org/wiki/Computable_analysis for some results in recovering analysis (limits, differentiation, integration, etc) using the computable numbers.
Many reals are constructible so they are accessible in the sense that it's a theory of phenomena I can act and build upon, and the theory discusses up to an infinite number of cases even if I won't build out that far.
Depending on how philosophical you want to get, it is not so clear what exists and what does not. Neither it is absolutely clear what means to be discovered.
I think that ideas can be discovered. In my opinion, theorems, or more precisely the proof of these theorems, for instance, are discovered. In the sense that they are not an invention, they "emerge" from more fundamental definitions.
You seem to be of the opinion that the only things that can be discovered are those that emerge from the "real world", whatever the real world is. That is, I guess, an acceptable interpretation, but I do not think is the only possibility.
To give a very concrete example, chess rules do not exist in the real world, but knight's tours are discovered.
On the contrary. I think that "real" and "imaginary" numbers are both discovered and neither concept has more reality than the other.
I think that even chess rules have a meaningful "existence" (in a Platonic sense) even if they are in some sense arbitrary. We can't discover "the one true rule system" but we can discover various possible systems and also theorems about them.
Negative numbers do of cause have their application in nature (negative charge etc.). But it is possible to do correct mathematics that have no interpretation in real live. E.g. 6 Students walk into an empty classroom, 10 walk out, you have -4 Students left.
> E.g. 6 Students walk into an empty classroom, 10 walk out, you have -4 Students left.
The only trouble here is in the assigned interpretation. You do have -4 students left compared to the initial state but there is no reason to assume the initial state was 0.
The personal expression can get very far too, further than perhaps an advanced alien race may know. Supposing an alien race never discovered category theory, their mathematics could still be more advanced than ours, but perhaps more verbose, longer to write down or otherwise bulky and inelegant. The question of whether elegant mathematics is more advanced I guess becomes tricky to answer. But pedagogically and for practical application, elegance is paramount.
I think the most celebrated mathematicians are such because of their personal expression. And that "style element" also inspires people to go further than before. Grothendieck, who featured here a few weeks ago, is a case in point.
> It's silly to think that the sqaure root of negative one is something real to be discovered.
As you said, the complex numbers are the unique (up to isomorphism) algebraic closure of R. Given that they arise as the unique solution to (in my opinion) a pretty fundamental question about the real numbers, I think it's fair to say that they've been "discovered", not "invented".
Real numbers might be less real than you think. Once you get beyond the number of possible quantum states in the universe, what do higher numbers really mean?
1. Isn't an infinite universe consistent with observations?
2. Yes, real numbers, much like complex numbers, were "invented". But complex numbers lay hidden, waiting to be discovered, as the algebraic closure of the reals, and similarly, the reals can be discovered from the "simpler" ideas of ordered field and Dedekind-completeness.
3. That's the weird thing about mathematics --- when you invent things, you leave a world of discoveries for others to make, and sometimes those discoveries are that your invention has inside it a perfect mirror image of another invention.
4. Once you leave classical logic, you suddenly have a lot more room for invention, because there are several competing definitions of "real number", and none is definitively better.
It might be expressed differently, but the concept of addition seems very fundemental, and additive inverses (negative numbers) are a very real part of addition.
> It's silly to think that the sqaure root of negative one is something real to be discovered
If we assume the aliens also have a need for multiplication, combining it with multiplication we get polynomials, and to factor polynomials you need complex numbers.
Besides, if they think of quantum mechanics they'll surely need some way to express what we call the complex numbers too.
Even if they don't think of them as "numbers", they'll absolutely discover analogous concepts and theorems.
"They have the concept of "minus", which is just another way of adding with a negative."
You have chosen to generalize it into the anstract concept of negative numbers. That doesn't mean that negative numbers are a fundamental truth about nature to be discovered.
Think about it like functional programming. Is that discovering some natural law of computing? Or is it inventing a new way to express computation? Of course it must be the latter, because you can compute anything without the use of functions.
Are you sure negative numbers are not "real" in the same way natural numbers are?
That would mean particles like electrons with a negative charge are not "real", too. And someone can argue that his negative bank balance is not "real".
As for complex numbers, I fail to see how they are less real than real numbers. Real numbers are points on the real axis, while complex numbers are points in R^2 plane. If we question the reality of real numbers, we can also question the reality of the plane.
While negatively charged particles are real, the 'negative' charge is a way to distinguish them from a 'positive' charge. It's a naming convention more than it is an actual negative value.
We could easily call them 'black' and 'white' particles and the system itself would stay the same
This is not true. You're losing a crucial property in your translation, namely that the negative numbers are the additive inverse of the positive numbers. In other words, it's essential for electrodynamics that a + (-a) = 0. You could add this requirement to you 'black' and 'white' terminology, but then you've just arrived at negative numbers under a different name.
Don't let the naming confuse you. The name "negative" might be a peculiar, human-specific thing, but the concepts of addition and additive inverses are not.
The etymology of invent has the terms 'contrived' and 'discover' baked in.
if we take the contrived root, rather than just dead-ending to say that invent is synonymous with discover,
we find that it is rooted in the ability to 'compare' and 'imagine'.
From this we can then formulate the opening statement.
I am not sure of the implications. Does it even matter if Math is invented or discovered? Maybe it's both, and it isn't a contradiction between invention and discovering?
We describe things having a certain radiation wavelength as having the color yellow. In that sense, yellow it's an invention. That doesn't mean the radiation doesn't exist.
But to complicate things a bit, some things don't exist unless we observe them. This is the case with states of particles described by quantum mechanics.
Math is more than a science, is the sciences upon which most other sciences and tech are founded. You can model anything in a computer and running on a computer using math. You can describe logic, natural language, technology, biology using math.
In that sense, being the building block of other sciences, math is more akin to a language. Two physicist use math in almost the same way two people use English to describe things and communicate ideas.
But math has building blocks, too. Set of axioms upon which any mathematical object and theory can be constructed. The most popular as of now is Zermelo set theory. There are more such fundamental theories, sometimes very different between them.
So, to see if Math is discovered or invented, the easy thing to do is to see if a set of axioms can be discovered or is invented.
> Does it even matter if Math is invented or discovered?
It matters for philosophical inquiry. Does philosophy matter? It matters for people who find it valuable, the same way art, music or literature matters.
You think logic, the art of making distinctions, the study of critical thinking and the foundation of mathematics only matter in the same way art does?
Ridiculous hubris, to claim to speak on their behalf, to claim to know the opinion of so many and diverse minds, and in such a crude misdirection to boot.
I agree 100% with everything you say. However, I don’t believe that the “impact on lives” of scientific advances, whilst certainly being of tremendous interest and importance to the humans, carries into much universal significance.
If you buy into Wittgenstein's rule-following paradox.
You could say anything about anything.
> no course of action could be determined by a rule, because any course of action can be made out to accord with the rule
If I had my own way of (mis?)interpreting him, I think he was alluding to what we now call "strict evaluation" in Programming Language Theory. A diligent rule-follower.
My question asking if you value products of philosophy that have become essential to our society neither contains an unjustified assumption and thus is not a loaded question and still stands and thus has not backfired.
It contains an unjustified assumption that “logic, the art of making distinctions, the study of critical thinking and the foundation of mathematics" and [other?] art belong in different categories.
It also takes an incomplete view on "critical thinking". Drawing distinctions is complemented by abstracting similarities.
The creation of knowledge (in all its forms) is itself a form of artistic self-expression. It is essential to humans, and therefore essential to society.
As a programmer my medium of self-expression is software.
I am an artist as much as I am a logician and a scientist.
What I do is create. It also happens to be useful to others, which is why it pays fucking well too.
> the art of making distinctions, the study of critical thinking and the foundation of mathematics only matter in the same way art does?
A major movement (or several, depending on how you slice it) in modern philosophy takes aesthetics as first philosophy - so yes, arguably they matter exactly the same way art does.
I just think our quality of life would be much lower without the forementioned than without art. We'd be much worse off without any of them. My point is not to dismiss art but emphasize the utility of philosophy.
I think not the whole of logic, but this particular distinction -- what changes in the world if one is true versus the other? Is there even a difference that is manifest?
It's like the question of what underlies quantum mechanics -- is the Copenhagen interpretation correct? Everett? Some flavor of deBroglie-Bohm? If there's no way to tell the difference, does it matter?
It matters as far as the philosophy of it matters to you, but the concrete consequences of one way being true versus the other could very well be nil.
Copenhagen requires an extraneous assumption — which may turn out to have practical implications.
At the very least, we know it isn’t parsimonious, though that wasn’t clear until after it was developed (and the philosophical inclination to preserve locality was reasonable).
There is a distinction between logic, which is just the study of formal systems where you can prove theorems as in any other area of mathematics, and the philosophy of maths and particularly of mathematical logic. The same applies for set theory.
The first one can assert that (in classical first order logic) e.g. there are uncountable models of the natural numbers; this is irrefutable. The second one asks things like "is classical first-order logic even true and/or adequate in an epistemic sense" (this is different from the more pragmatic question of "is classical FOL useful for the problem at hand")? Similarly, something like Gödel's incompleteness theorems are unequivocally true but the question of what they "mean" deep down is nothing that really affects mathematicians' work in general.
Irrefutable and "unequivically" true if you take some classical FOL as a productive method of producing knowledge, a philosophical assumption. Philosophy came first, mathematical logic is just a formalization of that and the reason it was formalized at all and not somethimg else is epistemological.
Your entire argument attempting dismiss philosophy is philosophy.
I said that as soon as you fix some axioms (such as those of classical FOL), the conclusions are irrefutable. This is where mathematics begins. The question where those axioms come from or whether they are "true" are philosophical. The two disciplines are related, but separate.
Lots of mathematicians have different "foundational" beliefs from each other; some have studied or deeply thought about the philosophy, others may just speak to their intuition. However, this doesn't change the fact that they all come to the same conclusions from the same premises. E.g. a constructivist wouldn't be able to claim that a classical proof is "wrong", only that it's non-constructive and therefore unacceptable for some (philosophical or practical) reason; in fact non-constructive proofs can be seen as constructive proofs of some meaningless strings (e.g. the constructivist will maybe dispute the fact that there are discontinuous functions, but they will certainly accept the existence of a first-order derivation of the string representing "not all functions are continuous" from the axioms of set theory), the constructivist would just dispute that there is any meaning to these strings...
Philosophical implications are perhaps the most tangible implications there are: they shape the conception of reality and society that you live in. They only seem intangible because they are subtle and pervasive.
Patents cover inventions that are applications of knowledge. For example, chemistry is certainly discovered, but applying the use of a certain molecule as a glue or as a drug is very much patentable. Physics is also discovered, but a special lever system that is a direct application of Newtonian mechanics is patentable. The knowledge that a certain mathematical formula could be written and has certain properties is not patentable, but a device that employs that formula as an algorithm for, say, predicting stock prices, is (in certain countries at least).
I don't see how you are distinguishing between inventions and discoveries. The "discovery" of a mathematical proof is just an application of knowledge, why aren't they patentable? The "invention" of the light bulb, on the other hand, is just the discovery that putting various particles together in a certain configuration produces light in a predictable manner, why is this patentable? I don't see how a clear delineation between the two can be made.
I'm not. I'm saying that devices whose function is derived from mathematics (algorithms) have the same status as devices that derive their function from physics or chemistry. In either case, "application" doesn't mean some intellectual use but commercial use. The purpose of patents is to protect commercial applications in exchange for sharing the knowledge that led to their creation. The knowledge itself is never protected.
When the lightbulb was patented, everyone was free to learn, study, and disseminate the physical knowledge of how a lightbulb works. What you couldn't do is build one and sell it. Whether mathematics is invented or discovered, the knowledge itself is never protected, but if some non-obvious algorithm has a commercial application, it can be patented so to protect building devices that apply that knowledge for commercial use.
But that's the whole point, patents are awarded to things deemed inventions not things deemed discoveries. If you can't delineate between the two you can't say what is and what is not patentable.
Fourier analysis couldn't have been patented, despite numerous commercial applications. It's just the patent system is setup to only reward low-level innovation, so it arbitrarily excludes research level innovation by terming it discovery.
Yes, what is patented is some commercial application of some knowledge, and that application is the invention. It doesn't matter whether the knowledge itself is invented or discovered. When an algorithm is patented, we're no more patenting math than we're patenting Newton's laws when a car brake is invented. We're patenting a commercial application of either, and that must be the invention.
Commercial application is irrelevant to a patent law. Patent applications need only demonstrate "eligibility, utility, novelty, and non-obviousness." Math and other basic research are ruled out on eligibility grounds, not for being non-commercial, but rather for being "abstract ideas."
Also, algorithms are a bad example, as they are just mathematical functions, i.e. "abstract ideas." They should not be patentable. Although I realize patent law is not actually logically consistent.
Commercial application is the motivation behind patent law, and logical consistency is not the point, but rather legal consistency. When you patent an algorithm you do not patent the idea any more than you patent thermodynamics when you patent an engine. You patent a particular application of an idea that performs some function or functions. The algorithm -- or thermodynamics -- stay completely free for anyone to know, study and disseminate. In fact, patents are designed to make the knowledge public so that they could be studied and improved upon.
Whether or not it works is a separate question (and I completely agree that software patents do not perform their role), but patenting a device to predict stock prices is not patenting math, just as patenting a drug is not patenting chemistry.
It matters because what if you can't actually describe the underlying foundations of the universe and reality with absolute accuracy as a mathematical equation. We get very close, but if we're always off even by ten to the minus whatever, have we really 'solved' it.
Maybe there is some other conceptual framework that we've not or are not able to cognitively express that underpins things.
to see if Math is discovered or invented, the easy thing to do is to see if a set of axioms can be discovered or is invented
A math theory arises from the axioms it is based on. You just rephrased the question and added the word "easy".
Put it another way, starting from a set of axioms we get a simple ( by some semiobjective definition of simple ) pure math theory that predicts reality to a rediculous level of precision.
Those initial axioms, were they discovered or invented?
Math is only sometimes done that way. Often an interesting field gets axiomatized later. Calculus, arithmetic, geometry, algebra, all existed productively for centuries before axiomatization.
Same way you can build a programming language without a formal spec. Yes, you might find an ambiguous piece of code later, or paint yourself into a corner. But you can do quite a lot without axioms.
Could the same be said about just about everything?
Like Music. Is a Song just a combination of notes, beats, intervals, voice etc. waiting to be discovered? Or is it something that a musician invents in her brain through talent, experience, practice and trail and error.
Or for that matter, a startup idea? A product/service that would bring immense value to its consumers, but it is not there yet, waiting to be discovered.
I guess philosophers must have dwelled on such questions before.
I think I get your point and I‘m open to ideas which drive home your point in a more precise way. However, the main difference between concerning ourselves with whether or not mathematical objects exists vs music or start-up ideas is their place in building a foundation for understanding reality. Science has been a productive program for understanding reality. It happens to be underpinned by math. And as a result, we have encountered some pretty interesting relationships between what the math tells us and what we observe. For instance, neutrinos and black holes were known to exist mathematically before they were ever observed and measured “in reality”. It’s not clear to me that music and start-up ideas hold up to math when it comes to being a tool for understanding or describing our reality, or perception the rig. Math is somehow fundamental to many successful enterprises which seek to describe and explain reality. Therefore, it becomes the point of focus of much philosophical inquiry when seeking to understand reality. Hence the foundational question, do mathematical objects exist? And not, does music or blockchain kittens exist?
Yes, they do, in an abstract sense. Existence in math means a very different thing from existence in physics. There are mathematical objects that exist, and those that cannot and do not exist. Someone posted a SEP entry earlier, which is a good start on this topic.
>the main difference between concerning ourselves with whether or not mathematical objects exists vs music
There is no 'vs', because there is no difference. Music is math, any song or sound is a mathematical object. A physical waveform that you hear can be encoded digitally in numbers: ones and zeros. So any given wav/flac file is just a bunch of numbers that give rise to the qualitative experience of sound, when interpreted in a certain way.
For example, a digital waveform consists of samples, each sample takes 16 bits to encode. Sampling rate of 44.1kHz is 44,100 samples per second. So you have 16 bits per sample x 44100 samples per second per channel x 2 channels x 300 seconds = 2^423,360,000 possible permutations of a 5-minute audio file without compression, which is a number with over 127 million digits. A little percentage of these permutations would count as music (even if your tastes are really diversified), most of it would just be noise. But all these possible 5-minute audio files include not only every song and every performance that existed or will exist. They also include every possible sound recording: songs that will not be written, Paul Graham saying that he hates HN, Paul Graham saying that he loves me and the rest of the file is silence, you and me discussing this topic with Plato for 5 minutes, etc, etc. The data exists and can be discovered and listened to, even though some of these examples are obviously not physically possible (i.e. Plato is dead).
So all music already exists mathematically, and it can be a useful mindset that your job is to discover it. A lot of musicians see it that way, Tessa Violet, for example:
https://youtu.be/QzBoGVToWEo?t=342
You can try to answer a simpler question: does the number 3 exist? If you define 3 as "the obvious common feature shared by three cows, three pencils, and three stars in the sky," then the question becomes "does this common feature exist?" and the answer, then, is, well, of course, yes, it does - because it's obviously there!
Think of it this way: Anyone can invent a new song, and no matter how derivative or lacking in artistic merit, they can still claim to have composed a new piece of music.(La-la-la-laaa. There, I just did it! Short but sweet.)
But you cannot just "invent" a new proof to the Pythagorean theorem. Any proof that a^2 + b^2 = c^2 will have to clear the hurdle of being demonstrably "true."
I'll never understand how someone otherwise so apparently intelligent can be so religious. Weird, the systems that we spent massive amounts of energy designing to precisely describe reality do that better than all the ones that we threw out along the way!
> I'll never understand how someone otherwise so apparently intelligent can be so religious.
Why not? Religious people believe that God is the Most Wise. Mathematics in nature attest to that attribute of God (as well as to other attributes of Him). An intelligent and religious person would recognize that, knowing it is God who came up with all the rules that keep the universe in balance.
I'll never understand why some people believe science and creationism can't go hand in hand.
If you want to understand why some people believe that science and creationism don't go hand in hand, then studying epistemology would help you do that. Most introductory texts will cover what you need.
Sounds like I got something new to learn during this lockdown, thank you. After a quick look at the wikipedia page on epistemology (very interesting), I'm not sure how this explains why some people believe that science and creationism don't go hand in hand.
A super super short version is about the question "how do you know X?" Its generally (although not universally) agreed that there are two types of knowledge, a priori and a posteriori. Roughly, in the first case, all you can deduce is implications from assumptions. For example, "if X and Y are true then Z is true." Such things don't require experience. The second case requires evidence gained from experience. Mathematical knowledge is of the first type, scientific knowledge is of the second type. The school of rationalism prioritizes the first type and the school of empericism emphasizes the second. That the universe was created by an omnipotent omniscient and benevolent god is not an emperical fact but requires some kind of a priori supports. Thats why you see so called proofs of god rather than scientific theories of god. There are other options of couree like mysticism, the claim that knowledge can be gained outside of these methods, or an emphasis on faith, that we don't take these things to be justifiable.
I probably butchered this but it gives you some starting points to research and hopefully shows a model of the debate. Huge topic. Check out https://plato.stanford.edu/entries/epistemology/ as a hardcore crashcourse or check "crash course philosophy" on youtube for the relavent sections at a more HS level. SEP has entries on god, faith and mysticism as well as rationalism and empericism too.
Careful with your use of the word creationism. I guess you mean it in the sense of the very vague and general claim that “god created the universe”, but in my experience it is used to refer to a much more specific set of claims, such as “the Earth is about 6000 years old”, which is absolutely not compatible with science.
According to Wikipedia: 'Creationism is the religious belief that nature, and aspects such as the universe, Earth, life, and humans, originated with supernatural acts of divine creation.'.
I choose my words carefully. I get what you're saying though, but I think that's more of a subjective matter.
And who/what created God -- e.g. who is God's God? And so on? It seems like humans trying to understand a hypothetical God are not unlike ants trying to understand quantum physics.
Why is there a need to insult anyone when it is you yourself that lacks understanding of the matter. We believers believe that God was uncreated, He is unlike His creation. Only created things have a creator. And God created time itself, so there is no before or after Him. He was always there. So there's no other answer than "that's impossible" to your question.
Of course this is hard to understand, but what would you expect? We can't even comprehend the tiny amount of His wisdom He has given us, let alone comprehend all of it.
If you're in the sahara desert you basically have all materials you need to create an iPhone. Not in a million billion years will there ever come an iPhone into existence by mere accident of natural forces in that desert, would there? Yet you believe that living things, which are not even comparable in complexity to an iPhone, were formed by a chain of coincidental events in the universe?
You can see God if you are willing to, unfortunately most atheists keep their hearts closed...
You ascribe beliefs to me via stereotype. I don't "believe" in the Big Bang Theory. I'm content knowing that I don't know. It doesn't strike me as rational to speak in such absolutes about subjects which are beyond human comprehension. To each their own.
> It seems like humans trying to understand a hypothetical God are not unlike ants trying to understand quantum physics.
> You ascribe beliefs to me via stereotype.
Not that I feel offended by the first quote, since I don't believe in a hypothetical god, but how am I the one stereotyping you? If you don't believe in God, which you made clear in your first comment, then you disbelieve in Him. So that means you're an atheist, right?
Or am I misunderstanding the concept of being an atheist? To each their own of course.
> If you're in the sahara desert you basically have all materials you need to create an iPhone. Not in a million billion years will there ever come an iPhone into existence by mere accident of natural forces in that desert, would there?
The evidence says that actually did happen. Abiogenesis and the evolution of human beings were just part of the process of producing an iPhone through a "mere accident" of natural forces over the course of the last 4.5 billion years or so. Producing an iPhone was never the goal, of course–natural processes don't have goals. But it was one of the consequences.
If God can exist without being created, so can what you refer to as "His creation". This idea that the visible universe requires a Creator, who in turn does not require a Creator, is arbitrary and capricious.
You can invent and pick axioms in many ways that (probably) won't lead to inconsistencies.
But they won't all be powerful enough or relevant in the real world.
Well, then it sounds like your reasoning gets it backwards: the axioms that produce systems without significant consequences or connections outside of their own abstract realm end up being ignored.
Or in other words: the constraints on maths are imposed from outside of maths.
Doesn't this imply that, while you can invent all the axioms you like, you must discover which ones are consistent with each other and with experimental results.
Arithmetic existed long before its axiomatization. Arithmetic was useful and no one stumbled upon contradictions in it. So it was natural to suppose that it can be described by some axiomatic system. Peano found it.
It is a system for modeling concepts invented by man. Everything that falls out of such a system is a product of the invention. Numbers don't inherently exist. Everything derived from that concept can't be a "discovery".
This is an excellent point. When I took algebra as an undergraduate I was blown away by the fact that you can choose any axioms and then derive an algebra based on those axioms. I was blown away because prior to that course I just assumed that our “standard” axioms were immutable.
Can you give some examples of axioms in pure math that run completely counter to our physical world? For example:
It is NOT possible to draw a straight line from any point to any other point.
It is NOT possible to extend a line segment continuously in both directions.
etc...
or
Things which are equal to the same thing are NOT equal to one another.
If equals are added to equals, the wholes are NOT equal.
The whole is LESS than the part.
Note that the original forms of the above axioms "make sense" to us because everything in our physical experience agrees with them. So when you said that the "physical counterpart ... is immaterial", I was curious to see an example of a "physically impossible" axiom.
> If you only look at mathematics I think it's simply: - Axioms are invented - Conclusions are discovered
How would you revise this statement if we lived in a "Mathematical Universe", like Max Tegmark's hypothesis.
> The magic part for me is that some axioms have been chosen so well that their conclusions are confirmed in the real world.
It's actually hard to avoid Turing completeness, and once you have that, any recursively enumerable function is calculable. All you need is addition and multiplication on numbers.
If B follows from A via logic, but A is invented, isn't B invented as well then?
Just to take a recent example which was mentioned here, Geometric Algebra[1]. There you assume you have some objects which aren't numbers but which when squared equals a given number. By doing that a bunch of nice results have been discovered.
However to me the basic premise, take some objects which aren't numbers but which square to a number, feels very much like an invention. So as such wouldn't the nice results be inventions as well?
> If B follows from A via logic, but A is invented, isn't B invented as well then?
Not really, because you have no choice about B - it's true (or rather it follows from A) whether you want it to or not. You could have picked a different A, but once you picked A then B was fixed.
People invented the electric chair because people discovered that high-voltage electricity can kill people. You might invent a particular axiom because it has an interesting mathematical consequence (e.g. the axiom of choice was invented to allow Hilbert's basis theorem to be proven), but what you invented was the axiom, not the consequence.
It sounds to me like you are giving special precedence to "numbers" in your considerations, as well as drawing your intuition for "square" from working with the integers or the reals. A square is just the result of a binary (product) operation where both of the operands are equal. The operation can be defined on an algebraic object with much richer structure than the integers or reals, and the operation itself can be much more complex than, say, integer multiplication. So the geometric product of geometric algebra is just one animal in the zoo of examples. You might call the specifics of the geometric product operation an "invention," but not because the square of a non-number can be a number.
Well if one would consider numbers and their product "discovered", then my point was that introducing the objects which have the property that they are not numbers but square to numbers seems to me like an invention.
At least if there is to be any meaningful distinction between a discovery and an invention.
Mathematics is just a language and likely a human discovery method for rules that encapsulate the universe and cascade down. A method invented to discover discoverable rules.
You can build all sorts of bizarre structures in ZFC that seem to have no relationship with reality at all. You can prove that almost all real numbers can't be computed or even described. There is an entire tower of infinities bigger than the natural numbers that telescope up and up. The axiom of choice gives you all sorts of nonphysical results.
The world of ZFC doesn't feel very constrained by the actual universe, is all.
And if you don't like ZFC, you can pick another set of axioms entirely- there are several alternatives.
Sure, math/ZFC can describe potential and impossible universes including ours. But the motiviation for its invention as a language was very likely to understand relationships in our reality, physics, trade etc.
Math is purely abstract science yet it describes the world around us to the utmost precision. Does that mean that this world is simply ... a math model which means everything around us is ... not real?
I've long stopped believing in free will because everything points at it being an illusion of our brain because we're a product of this world and we had no chance of influencing the conditions which brought us to life, and even after our minds and consciousnesses form it's hard to believe they are fully autonomous and not simply a function of the processess in our brains we're simply not aware of.
If you think about all of it, it becomes utterly depressing as you begin to realize you're a biological robot, a byproduct of the universe evolution which couldn't care less about our species and this little tiny blue planet.
> Math is purely abstract science yet it describes the world around us to the utmost precision. Does that mean that this world is simply ... a math model which means everything around us is ... not real?
You can only reach this conclusion if you categorize math as "not real". If everything we interact with is a perfect mathematical object, I prefer to take that as evidence that all mathematical objects may be real, however abstract they seem.
What’s depressing about being a nanotchmological robot attempting to conquer the solar system and spread his technology and nanotechnology? It sounds rather exciting and fascinating. Perhaps if the depression is too severe you need to check out your programming ;-)
Mathematics has not been static throughout history. It has evolved at the same pace our science and discoveries have evolved, and as such it has incorporated all necessary elements to help describe new phenomena. To the point that now is able to describe never observed phenomena.
So no, our world is not unreal because it can be described with maths. Maths have evolved to describe it. Same as human language has evolved to support our close environment and our human interactions.
> Math ... describes the world around us to the utmost precision.
It does no such thing. The models we create using mathematical language do.
Math is a precise language that can be used to describe the world around us precisely. It can also be used to describe utter nonsense or utter fantasy precisely.
Yep, the counterpoint to the unreasonable effectiveness of maths in the natural sciences is accompanied by the unreasonable ineffectiveness of maths in the social sciences, humanities, etc. Although at least to some extent statistics is the one branch of mathematics that does seem to be applicable (but the way this influences results, as opposed to methods, seems to be mathematically not that interesting).
There have been a lot of efforts of modelling social phenomena, the arts, etc. mathematically and while there are some interesting partial results (e.g. that languages can to some reasonable extent be analysed by parse trees or some aspects of music), most "grander" theories I have seen do not really stand up to much scrutiny.
I'm not an expert, but I could assume that part of this is due to a lot of nonlinearity in the phenomena studies, which means that many classical methods don't work well; maybe chaos theory etc. could shed more light on these things, but I don't know enough about it.
> Yep, the counterpoint to the unreasonable effectiveness of maths in the natural sciences is accompanied by the unreasonable ineffectiveness of maths in the social sciences, humanities, etc.
Except it's not ineffective at all in these subjects. Social science experiments have so many variables that the small experiments that can be conducted given the financial resources are insufficient to infer a good model. This is not a math problem, it's a money problem.
If it's ineffective in practice, I consider it to be ineffective.
But even if you could design an experiment perfectly and come up with some strong statistical evidence and then had other means to tease out what is actually causal and what is only correlational, you'd only know what influences what and not necessarily why. But yes, I did say that statistics/probability is some rare exception.
You can study group theory and understand the way physical forces work better, or hilbert spaces to understand quantum mechanics, but I haven't yet heard of anyone who has studied topology or galois theory and found that incredibly useful for understanding social phenomena better.
> If it's ineffective in practice, I consider it to be ineffective.
Except you have no way to conclude that it's ineffective. Analytical solutions require data to study. Without data, or with little data, what analysis are you going to perform? At best, broad statistical correlations, which is exactly what we find.
You're effectively claiming that spoons are ineffective at a restaurant that provides only forks. Well no, if a spoon were available, then it would probably work just fine.
> but I haven't yet heard of anyone who has studied topology or galois theory and found that incredibly useful for understanding social phenomena better.
The main stumbling block is that mathematicians are interested in mathematical problems, and so they make a common but mistaken assumption that social sciences either don't have such problems, or they are too messy for elegant math. Take it from a mathematician, this is incorrect: https://www.mathtube.org/sites/default/files/lecture-notes/S...
You may have a point. And I did forget about things like social choice theory or game theory (although I'd assume that partially this is also due to e.g. social choice procedures or "games" often being very limited and artificial settings where by their very nature the relevant space of options/outcomes can be explored in some systematic way, which is generally less the case in more organic, complex settings, such as e.g. gradual societal changes).
When it comes to economics, I know that a lot of people don't agree with the basis of many mathematical models that are used, but I'm not an economist, so I can't speak to that.
So maybe I was overzealous in discounting mathematics for the social sciences altogether. Still, I would contend (albeit with much less evidence):
- There's some measure of people trying to construe "nice models" of things in those subjects instead of trying to make sure they agree with reality. I have a degree in linguistics and I've seen this over and over. Most of these models haven't convinced me at all.
- The amount of maths that you either need or at least benefit from in order to do good research in such areas is still substantially lower than in, say, physics. I think it's still to be noted how we can describe much of physics just with a small number of economic and elegant models. I haven't seen anything comparable in any of the social sciences.
> If you think about all of it, it becomes utterly depressing as you begin to realize you're a biological robot, a byproduct of the universe evolution which couldn't care less about our species and this little tiny blue planet.
It can also be liberating and uplifting. You're a part of the universe - you are the universe expressing itself. The result of uncountable generations of brutal evolution, you're quite a miracle!
>I've long stopped believing in free will because everything points at it being an illusion of our brain because we're a product of this world and we had no chance of influencing the conditions which brought us to life
Free will doesn't depend on the universe being deterministic or not. Things that haven't happened yet are likely going to happen in a certain way based on the trajectory, but it doesn't mean that you can't change your own path in a meaningful way, you're free to take a harder path vs. an easier or more obvious one.
>but it doesn't mean that you can't change your own path in a meaningful way, you're free to take a harder path vs. an easier or more obvious one.
This is of course debatable, there is some evidence to suggest that the feeling that you took a certain path is illusionary. Your brain made the decision, then you became cognizant of the options and you felt yourself making the decision[0], but if you could rewind the universe, you would always choose the same path based on your 'brain-state' at that time. At least that is what I understand of the most extreme "free-will doesn't exist" position.
Just got lost in the SEP rabbithole [1], still not sure where I land on this issue.
[0] : like how you feel thoughts 'bubble up' during meditation, you didn't actively 'think' those, they appeared to you. EDIT: atleast that is what it feels like
That things would repeat after a rewind, given the same "universal state" just contradicts free will from the perspective of looking backwards (to the then present), not in the current present or looking into the uncertain future.
The platonic universe of the entirety of math sounds (this week at least) like Wolfram’s “Universe of Computational reality” of which the physical reality is just a subset.
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[ 0.85 ms ] story [ 502 ms ] threadIn intuitionist mathematics there is only potential infinity, no actual infinity. Constructive set theory differs from Zermelo set theory.
That has many consequences in practice. Applying intuitionist mathematics to physics we can come to the conclusion that time flows and it helps reconcile quantum mechanics with general relativity.[1]
[0] https://en.wikipedia.org/wiki/Intuitionism
[1] https://www.quantamagazine.org/does-time-really-flow-new-clu...
Can you expound on this a little more? I'm completely new to the idea of intuitionistic mathematics and much more so to its applications in physics; the constructive approach to thinking about objects and properties is very refreshing and I'd like to hear how you've related those principles with the paradox of time in the context of classical physics.
Unfortunately no. I got the info from Quanta Magazine article. I am not knowledgeable enough to respond on this. Even though I did some high level math courses in University, I wasn't interested enough.
I am sure other HN users understand this stuff better.
And Im wodering why Im not looking more into the philosophy of math, i find this interesting because I am mainly an intuition driven person. Any ideas for courses online?
But remember that (as far as I know) you can do intuitionist mathematics in classical mathematics, but not the other way around. So you can think of intuitionist mathematics as being embedded in classical mathematics.
The second paragraph is somewhat true --- at its base, the modern use of the term "intuitionistic logic" refers merely to not accepting the law of the excluded middle. However, there are varieties of intuitionism that are very, very inconsistent with classical logic, like those adopting the idea that "every function between real numbers is continuous".
For example, any topos has internal logic on subobjects of at least a Heyting algebra. In some cases you specialise to a Boolean algebra.
However, when you write things down such as your arguments around functors and constructions, you argue according to classical mathematics externally. When you enter into the category (which is chosen not to be Boolean) and argue on the subobject structure, then you are entering intuitionistic logic. This is what I mean when I say that classical logic can be argued to have intuitionistic logic embedded in it.
The cardinality of the reals being equal to the cardinality of the naturals is then something you have to construct inside the topos. So you need to construct along the spirit of a natural numbers object (perhaps a real numbers object) and then use the internal logic to argue about continuity.
I checked the paper that I was thinking about and the condition that I saw there is distributivity being dropped rather than double negation being dropped.
> you can do intuitionist mathematics in classical mathematics
There are a number of embeddings of classical logic into intuitionistic logic. The most popular/obvious is the double-negation embedding that maps a proposition to its double negation.
What you are referring to as a proposition is an element of the Heyting algebra or the Boolean algebra.
My point is something else: I don't know of a way to have a intuitionistic exterior logic and a classical internal logic.
Though I'd like to think that most species would come up with some version of calculus, even though the notation will be obviously different. Afterall, two of our own species did so independently.
Location was not specified ...
https://jeb.biologists.org/content/210/2/198
Can you get data on this from a Monte Carlo method of study on match play between sighted and blind players playing go and chess at distance and blind which requires a mental model of the board state and translations?
http://www.math.pitt.edu/~bard/bardware/classes/0220/dkc.pdf
How can you be sure?
It is not surprising at all that almost all blind mathematicians are geometers. The spatial intuition that sighted people have is based on the image of the world that is projected on their retinas; thus it is a two (and not three) dimensional image that is analysed in the brain of a sighted person. A blind person’s spatial intuition on the other hand, is primarily the result of tile and operational experience. It is also deeper – in the literal as well as the metaphorical sense. […]
recent biomathematical studies have shown that the deepest mathematical structures, such as topological structures, are innate, whereas finer structures, such as linear structures are acquired. Thus, at first, the blind person who regains his sight does not distinguish a square from a circle: He only sees their topological equivalence. In contrast, he immediately sees that a torus is not a sphere […]
This is a quote from a book by Alexei Sossinsky, quoted here: https://onionesquereality.wordpress.com/2011/07/31/blind-geo...
Maybe visualization is more helpful, but you can visualize things without seeing.
A sense of depth and distance, or any kind of metric is helpful, and senses like vision and hearing might be helpful in forming that, but I'm not sure they are required.
From another point of view, math theories are congruent, and you can explain the same things without using geometry, but using algebra, or differential equations or number theory and so on. You certainly can explain any math concept using just modern set theory - maybe it is not the most convenient thing to do.
Except once axioms have been fixed, the resulting geometry will be the same wether you perceive reality via visible light or echolocation.
https://corecursive.com/035-bartosz-milewski-category-theory...
But, planar geometry (primarily involving triangles and squares) does not actually exist in nature. Which means that planar geometry is an approximation technique developed by the brain in order to begin to understand the actual much more complex geometries that appear in the real world.
This also suggests that mathematics is 100% constructed by the human brain, even if it is highly-influenced by relationships found in the physical world.
But, this makes sense because we really don't ask the same question about human language. We almost never ask: is human language constructed or discovered?
As I see it, mathematics is both discovered and invented.
We can model every existing thing in every possible world using math. Even if both the set of all things that might exist and all possible mathematical constructions are infinite, the later is larger. That's because we can also construct mathematical models of things that doesn't exist.
So it looks that from the set of all possible mathematical constructions, we extracted a subset that maps to objects in reality. That looks like a discovery process.
But we also constructed mathematical models of things that don't have corespondents in reality, so that much be more of an invention process.
Let's pretend for a bit that we forget all what we know and tomorrow we will start inventing things again. Or that a species of aliens start fresh on a planet.
The mathematic theories and notions we and the aliens might discover, build or invent might be different than the theories and notions we know today but would probably be equivalent. That suggests that mathematics just exists somewhere in its own world waiting to be discovered.
Why integers one might ask. For me, even rational numbers are such an approximation.
It is "created" because there is no guarantee that our efforts correspond to reality. In that sense we are just playing in the sandbox of what we can conceive.
The fact that mathematics corresponds so "unreasonably" to objective reality is because what we call objective reality is mediated by our brain's own idiosyncrasies and limitations of thinking.
It's no more surprising that external objective reality is mathematically predictable and describable than it is that our eyes can see shapes and some, but not all, light. That's the purpose of eyes and they tell us enough of what we need to know that we can survive.
Our brains are exactly the same thing- they tell us a story in a way which helps us to survive. Actual reality may be beyond our capacity to conceive of or worse, may seem like nonsense to us because it's aggressively illogical or specifically contradictory and reality therefore makes no "sense" to us.
But doesn't this dance kinda near the notion that I am a brain in a vat and none of you exist (read this question with me as the speaker or with yourself as the speaker)?
I would say our brains do have limits. I can't well envision a 4D object. But once we reduce things down to simple logical axioms and constructs, these exist as much as anything can be said to exist. Even if I was a simulation that didn't even have a brain, much less eyes, the concepts I come up with would exist more than the flesh I incorrectly thought I had.
I agree.
But this assumes logic and at least the principle of non-contradiction: not both A and not A - is how reality "really" is.
We can't get past the idea that it must be this way because only provable nonsense lies on the other side of this assumption. But our brains may be fundamentally unable to process ultimate reality and the nonsense a contradiction represents may be a statement not about reality but our brains, our thinking.
So logical contradictions aren't actually nonsense, they're the sound of us hitting the walls of what our minds can conceive of. All animal have such limits. We assume those limits look like darkness- stuff we can't peer into. What if they look like impossibility instead?
That's the point of view I'm entertaining here. I am not saying this is true. It certainly isn't useful or provable as far as I know, but it is possible.
The practical value of such an exercise, if it has any (and I think it does) is to twofold.
One seems to expand my imagination to the maximum extent pops me out of the assumptions that frame my thinking and this seeps into my thinking about things, technical problems, generally.
Two it confers humility and a certain openess and makes me less judgmental. So that, for example, when I hear or read people with claims to spiritual knowledge I don't automatically blow them off as crazy / bitter / ignorant because what they're saying "makes no sense".
Thinking and talking about Ultimate Reality capital U capital R, ought to fill us all with humility if we're being intellectually honest by our own standards. Yet, I find people totally lack that humility. They make huge pronouncements about Ultimate Reality which they can't really be sure of, and the effect this has on the world, and how we think of each other, and therefore how we treat each other, and even the effect on one's own mind, is one of diminishment generally.
Mea culpa, I was one of those people and I didn't like it.
If you get down to the core of knowledge and how we know something, this is what's really there, and it's good to be reminded of it.
Ctrl-F for "2D" (2-dimensional creatures), although I can't find the "discovery", maybe I remember it wrong.
Btw, this was one of the mindfucks I learned studying physics. Other ones are:
I could see the layman understanding of geometry being quite different than ours, limited by their own visual system. But their mathematicians and our mathematicians would discover the same math, through different paths and perhaps with different strengths and weaknesses and a different path of educating budding mathematicians from the grade school level to the post doc level.
Recursive functions in computer programming, or fractals in maths, are infinite but can be described. Each small part looks like the whole, but the whole can never be fully known within a finite universe. Is that "potential" or "actual" infinity?
I've been thinking a lot about infinity this past week, mostly because of Conway's lectures. I'm trying to figure out whether the following logic makes sense.
- Assuming that the universe is finite -> Infinity cannot be approximated with finite numbers (MIP* = RE) [1] -> Numbers cannot have infinite strings of digits -> The future can never be perfectly preordained (Gisin) [2] -> Free will exists (Conway) [3]
This is not a proof that free will exists! But I think this is a consistency proof that free will is finite.
Now you're telling me that there's another kind of infinity, and I want to know what that means for this thought.
[1] https://www.quantamagazine.org/landmark-computer-science-pro...
[2] https://www.quantamagazine.org/does-time-really-flow-new-clu...
[3] https://www.youtube.com/watch?v=tmx2tpcdKZY&feature=youtu.be...
Conway said that there either is determinism or free will. If the future can be perfectly preordained (Gisin) due to infinite possibilities, then there is no free will. But if there are finite options, I think that there is "free will" for particles and people to choose between those options.
https://youtu.be/tmx2tpcdKZY?t=1074
https://en.wikipedia.org/wiki/Free_will_theorem
There's no reason why universal determinism and free will need to be related. So trying to "prove" free will with math makes as much sense as "proving" free will with weather forecasting.
Even if that weren't true you still can't get to Conway's Step 3, because "free will" could just be statistical noise, and not willed in any sense at all.
Is it though? I came to the conclusion that one could consider free will the self-realization of a person in space-time -- in other words, free will is the only and only choice one can make (at each decision they face), that they make because of who they are thus far.
In that sense, "free will" is the same as the "person" (or the person's essense) -- and would it make sense for it to be any other way? Randomness wouldn't be free will, and having a second entity (e.g. a soul in dualism) do the decision making just moves the question and degree upwards (from "how the person has free will?" to "how the person's soul has free will?")
If a person faced with a choice A or B could go either way, then I wouldn't call that really free will (even though for some reason that's what most people have in mind when talking about the subject).
Either a person is a concrete personality/being and that manifests through its choices (which means that when faced with the A or B dillema they can only chose one or the other based on who they are), or they have some degree of randomness in their decision making (which makes them less of a person in my eyes -- randomness is not tied to our person, it is, as the name reveals, random).
When some people say "free will" they actually mean "free (independent) from the person". But that's not a will then -- a will denotes something being tied to a person (a person is their will).
A coin flip and a hand of cards are both random, but obviously cards are 'more random'.
This can be quantified - 'more random' is information complexity, or information entropy. (Two names for the same thing.)
If free will exists, then it must necessarily be a singularity of infinite information complexity, which is the same thing as a singularity of infinite randomness.
It's not, although many people think this. Consider that the majority of philosophers are actually Compatibilists, in which free will is compatible with determinism, and so they would disagree with your definition. If the majority of practicing experts disagrees with your view on their subject, it's probably time to revise your view.
Arthur Schopenhauer famously said, "Man can do what he wills but he cannot will what he wills." In other words, although an agent may often be free to act according to a motive, the nature of that motive is determined.
This position that the ‘free will’ people experience is the ability to ‘decide’ to take some and then take it, while then saying the the decision to do so was predetermined, is not what is meant (especially in the vernacular) by free will.
Therefore, while I will not support a metaphysical dualist basis for ‘free will’ it is clearly incorrect to dismiss an argument of such based on 1) an appeal to authority which is unsupportable (in that your claim about the position of a majority of ‘experts’ seems unsupported), and 2) that even if it was supported, that said appeal to authority is a valid refutation of an argument. Further, dismissal of an argument by choosing to substitute a contested definitional term (the meaning of free will) for one that is clearly not the same is not rhetorically valid.
Almost 60% Compatibilist, the remainder evenly split among three other options: https://philpapers.org/surveys/results.pl
> Arthur Schopenhauer famously said, "Man can do what he wills but he cannot will what he wills." In other words, although an agent may often be free to act according to a motive, the nature of that motive is determined.
You're assuming this is relevant. Turns out, it's not.
> Also, one of the primary criticisms of Compatiblism, dating back as early as Kant, is that what Compatibilists define as ‘free will’ is not free will as most people, including philosophers understand it.
Nobody definitively understands free will. Some people conjecture it has certain properties, mainly incompatibilists. So far they have mostly been wrong.
> This position that the ‘free will’ people experience is the ability to ‘decide’ to take some and then take it, while then saying the the decision to do so was predetermined, is not what is meant (especially in the vernacular) by free will.
Experimental philosohy suggests pretty definitively that people's moral reasoning agrees with Compatibilism: https://www.researchgate.net/publication/274892120_Why_Compa...
> that even if it was supported, that said appeal to authority is a valid refutation of an argument.
The other poster provided no argument, they just made an unsupported claim.
Eg, the “natural numbers” are a recipe to make as many as you want (by taking the successor of the last one) but not an actual (as in existing) infinite collection — the only ones which exist are the ones you construct, and the “potential infinity” refers to the fact that you can always make a new one.
The "mental activity" part isn't strictly necessary. A constructive mathematics will produce results that are largely the same. What matters most is that mathematical objects are proven by construction, and so proofs have a direct translation to computation.
https://plato.stanford.edu/entries/philosophy-mathematics/
What struck me was what prof. Michael Merrifield said at the end of his explanation: https://www.youtube.com/watch?v=CiHN0ZWE5bk
On the question about what is the reality, what explanation is the true one, Merrifield said that the Math works on all of the explanations and what those explanations do is simply to model the behaviour of nature and not necessarily reflect the reality.
That's how Newtonian physics and relativistic physics are both correct models, tools to model nature and simply can be used to whenever suitable.
Wouldn't that mean that mathematics is just an invented tool to reason about physical models?
I think that this can be seen in the fact that sometimes mathematicians and especially physicists can reason about objects that they are not sure about what the right definition should be. Many mathematicians reasoned about continuous functions long before we had a concrete definition of them. But when Cauchy introduced the definition and Weierstrass proved it is equivalent to preserving limits (which was the intuition at the time), we had not truly discovered something new and mathematical.
This whole idea was then generalized to topology when it was shown that pre images of continuous functions preserve the "openess" of sets, i.e. we realized that no concept of distance was not needed to describe continuity, which is very surprising.
I am on a Penrose kick right now. Just downloaded The Emperor’s New Mind Audiobook.
It's silly to think that the sqaure root of negative one is something real to be discovered. It's just a stand-in to express complex relationships more succinctly. The same for negative numbers, for that matter.
It would be equally silly to think that the underlying relationships are invented. Nobody invented prime numbers, they were discovered.
If you throw away real numbers, then you lose major things in physics. I think for example you lose the wave function in QM.
It is not a question of whether they exist in nature, but rather whether they are the more superior technique, or not, to explain nature.
Complex numbers aren't "necessary for the continuum" as you put it, and some realists might argue that they don't hold the same "discoverability" as the reals.
I wouldn't. I think the reals and the complex numbers have the same "realness" and that neither represents any innate property of the physical world, despite how obviously useful they are in physical models.
Then again, have you ever observed a non-computable number as a component of the value of a wave function in nature? Real numbers add a lot of cruft which can never be practically observed (due to not being computable).
One of the main reasons we lose things is that our results have been built upon the real numbers since they were easier to conceptualize and to work with, but it's possible that a lot can be recovered (maybe even everything we care about) using only the computable numbers. For instance, see https://en.wikipedia.org/wiki/Computable_analysis for some results in recovering analysis (limits, differentiation, integration, etc) using the computable numbers.
I think that ideas can be discovered. In my opinion, theorems, or more precisely the proof of these theorems, for instance, are discovered. In the sense that they are not an invention, they "emerge" from more fundamental definitions.
You seem to be of the opinion that the only things that can be discovered are those that emerge from the "real world", whatever the real world is. That is, I guess, an acceptable interpretation, but I do not think is the only possibility.
To give a very concrete example, chess rules do not exist in the real world, but knight's tours are discovered.
I think that even chess rules have a meaningful "existence" (in a Platonic sense) even if they are in some sense arbitrary. We can't discover "the one true rule system" but we can discover various possible systems and also theorems about them.
Negative numbers do of cause have their application in nature (negative charge etc.). But it is possible to do correct mathematics that have no interpretation in real live. E.g. 6 Students walk into an empty classroom, 10 walk out, you have -4 Students left.
The only trouble here is in the assigned interpretation. You do have -4 students left compared to the initial state but there is no reason to assume the initial state was 0.
I think the most celebrated mathematicians are such because of their personal expression. And that "style element" also inspires people to go further than before. Grothendieck, who featured here a few weeks ago, is a case in point.
As you said, the complex numbers are the unique (up to isomorphism) algebraic closure of R. Given that they arise as the unique solution to (in my opinion) a pretty fundamental question about the real numbers, I think it's fair to say that they've been "discovered", not "invented".
2. Yes, real numbers, much like complex numbers, were "invented". But complex numbers lay hidden, waiting to be discovered, as the algebraic closure of the reals, and similarly, the reals can be discovered from the "simpler" ideas of ordered field and Dedekind-completeness.
3. That's the weird thing about mathematics --- when you invent things, you leave a world of discoveries for others to make, and sometimes those discoveries are that your invention has inside it a perfect mirror image of another invention.
4. Once you leave classical logic, you suddenly have a lot more room for invention, because there are several competing definitions of "real number", and none is definitively better.
It might be expressed differently, but the concept of addition seems very fundemental, and additive inverses (negative numbers) are a very real part of addition.
> It's silly to think that the sqaure root of negative one is something real to be discovered
If we assume the aliens also have a need for multiplication, combining it with multiplication we get polynomials, and to factor polynomials you need complex numbers.
Besides, if they think of quantum mechanics they'll surely need some way to express what we call the complex numbers too.
Even if they don't think of them as "numbers", they'll absolutely discover analogous concepts and theorems.
Inverses are just an abstraction that helps you rearrange operations.
You'd think something like accounting would need negative numbers, but nope, it gets along just fine without them.
Also deficits and surpluses seem like normal accounting terms.
You have chosen to generalize it into the anstract concept of negative numbers. That doesn't mean that negative numbers are a fundamental truth about nature to be discovered.
Think about it like functional programming. Is that discovering some natural law of computing? Or is it inventing a new way to express computation? Of course it must be the latter, because you can compute anything without the use of functions.
Are you sure negative numbers are not "real" in the same way natural numbers are?
That would mean particles like electrons with a negative charge are not "real", too. And someone can argue that his negative bank balance is not "real".
As for complex numbers, I fail to see how they are less real than real numbers. Real numbers are points on the real axis, while complex numbers are points in R^2 plane. If we question the reality of real numbers, we can also question the reality of the plane.
Mr Euclid would frown to hear that.
We could easily call them 'black' and 'white' particles and the system itself would stay the same
Don't let the naming confuse you. The name "negative" might be a peculiar, human-specific thing, but the concepts of addition and additive inverses are not.
The etymology of invent has the terms 'contrived' and 'discover' baked in. if we take the contrived root, rather than just dead-ending to say that invent is synonymous with discover, we find that it is rooted in the ability to 'compare' and 'imagine'. From this we can then formulate the opening statement.
We describe things having a certain radiation wavelength as having the color yellow. In that sense, yellow it's an invention. That doesn't mean the radiation doesn't exist.
But to complicate things a bit, some things don't exist unless we observe them. This is the case with states of particles described by quantum mechanics.
Math is more than a science, is the sciences upon which most other sciences and tech are founded. You can model anything in a computer and running on a computer using math. You can describe logic, natural language, technology, biology using math.
In that sense, being the building block of other sciences, math is more akin to a language. Two physicist use math in almost the same way two people use English to describe things and communicate ideas.
But math has building blocks, too. Set of axioms upon which any mathematical object and theory can be constructed. The most popular as of now is Zermelo set theory. There are more such fundamental theories, sometimes very different between them.
So, to see if Math is discovered or invented, the easy thing to do is to see if a set of axioms can be discovered or is invented.
It matters for philosophical inquiry. Does philosophy matter? It matters for people who find it valuable, the same way art, music or literature matters.
And if you believe art is about creativity, you can find beauty and art in in mathematics and other sciences.
Many great mathematicians were also art lovers, they appreciated music, they appreciated paintings and other forms of art.
I don't think there's a dichotomy between art and sciences.
The significance of the creation (if it has any at all) is not necessarily the concern of the creator.
Some people do Mathematics for its sheer beauty.
You could say anything about anything.
> no course of action could be determined by a rule, because any course of action can be made out to accord with the rule
If I had my own way of (mis?)interpreting him, I think he was alluding to what we now call "strict evaluation" in Programming Language Theory. A diligent rule-follower.
The significance of the creation (if it has any at all) is not necessarily the concern of the creator.
Some people do structural engineering for its sheer beauty.
The significance of the creation (if it has any at all) is not necessarily the concern of the creator.
Some people do [Scientific discoveries|Mathematics|Ethics|Political science] for its sheer beauty.
It backfired on you.
It contains an unjustified assumption that “logic, the art of making distinctions, the study of critical thinking and the foundation of mathematics" and [other?] art belong in different categories.
It also takes an incomplete view on "critical thinking". Drawing distinctions is complemented by abstracting similarities.
The creation of knowledge (in all its forms) is itself a form of artistic self-expression. It is essential to humans, and therefore essential to society.
As a programmer my medium of self-expression is software. I am an artist as much as I am a logician and a scientist.
What I do is create. It also happens to be useful to others, which is why it pays fucking well too.
> It matters for people who find it valuable, the same way art, music or literature matters.
Key phrase "the same way".
There's nothing wrong with being wrong, yet you strike me as a person who doesn't like it.
A major movement (or several, depending on how you slice it) in modern philosophy takes aesthetics as first philosophy - so yes, arguably they matter exactly the same way art does.
It's like the question of what underlies quantum mechanics -- is the Copenhagen interpretation correct? Everett? Some flavor of deBroglie-Bohm? If there's no way to tell the difference, does it matter?
It matters as far as the philosophy of it matters to you, but the concrete consequences of one way being true versus the other could very well be nil.
At the very least, we know it isn’t parsimonious, though that wasn’t clear until after it was developed (and the philosophical inclination to preserve locality was reasonable).
The first one can assert that (in classical first order logic) e.g. there are uncountable models of the natural numbers; this is irrefutable. The second one asks things like "is classical first-order logic even true and/or adequate in an epistemic sense" (this is different from the more pragmatic question of "is classical FOL useful for the problem at hand")? Similarly, something like Gödel's incompleteness theorems are unequivocally true but the question of what they "mean" deep down is nothing that really affects mathematicians' work in general.
Your entire argument attempting dismiss philosophy is philosophy.
I said that as soon as you fix some axioms (such as those of classical FOL), the conclusions are irrefutable. This is where mathematics begins. The question where those axioms come from or whether they are "true" are philosophical. The two disciplines are related, but separate.
Lots of mathematicians have different "foundational" beliefs from each other; some have studied or deeply thought about the philosophy, others may just speak to their intuition. However, this doesn't change the fact that they all come to the same conclusions from the same premises. E.g. a constructivist wouldn't be able to claim that a classical proof is "wrong", only that it's non-constructive and therefore unacceptable for some (philosophical or practical) reason; in fact non-constructive proofs can be seen as constructive proofs of some meaningless strings (e.g. the constructivist will maybe dispute the fact that there are discontinuous functions, but they will certainly accept the existence of a first-order derivation of the string representing "not all functions are continuous" from the axioms of set theory), the constructivist would just dispute that there is any meaning to these strings...
I was thinking more of tangible implications. Maybe there are some.
It impacts whether mathematics should be patentable, since patents generally covers inventions, but not discoveries.
When the lightbulb was patented, everyone was free to learn, study, and disseminate the physical knowledge of how a lightbulb works. What you couldn't do is build one and sell it. Whether mathematics is invented or discovered, the knowledge itself is never protected, but if some non-obvious algorithm has a commercial application, it can be patented so to protect building devices that apply that knowledge for commercial use.
Fourier analysis couldn't have been patented, despite numerous commercial applications. It's just the patent system is setup to only reward low-level innovation, so it arbitrarily excludes research level innovation by terming it discovery.
Also, algorithms are a bad example, as they are just mathematical functions, i.e. "abstract ideas." They should not be patentable. Although I realize patent law is not actually logically consistent.
Whether or not it works is a separate question (and I completely agree that software patents do not perform their role), but patenting a device to predict stock prices is not patenting math, just as patenting a drug is not patenting chemistry.
Is this intepretation of quantum mechanics still the canonical one?
https://en.wikipedia.org/wiki/Interpretations_of_quantum_mec...
I can't judge whether this proposal makes sense, but there's that.
0: https://arxiv.org/pdf/quant-ph/9510007.pdf
Maybe there is some other conceptual framework that we've not or are not able to cognitively express that underpins things.
A math theory arises from the axioms it is based on. You just rephrased the question and added the word "easy".
Put it another way, starting from a set of axioms we get a simple ( by some semiobjective definition of simple ) pure math theory that predicts reality to a rediculous level of precision.
Those initial axioms, were they discovered or invented?
Same way you can build a programming language without a formal spec. Yes, you might find an ambiguous piece of code later, or paint yourself into a corner. But you can do quite a lot without axioms.
and now, in the year 2020, have the axioms been invented or discovered?
I figure reality is part of math so if it was purely invented we wouldn't exist.
Where do Mathematicians look to discover Mathematics? In the depths of their own minds.
The part where you "look and think deeply" is discovery. The part where you "express your discovery in a coherent language" is invention.
Like Music. Is a Song just a combination of notes, beats, intervals, voice etc. waiting to be discovered? Or is it something that a musician invents in her brain through talent, experience, practice and trail and error.
Or for that matter, a startup idea? A product/service that would bring immense value to its consumers, but it is not there yet, waiting to be discovered.
I guess philosophers must have dwelled on such questions before.
Yes, they do, in an abstract sense. Existence in math means a very different thing from existence in physics. There are mathematical objects that exist, and those that cannot and do not exist. Someone posted a SEP entry earlier, which is a good start on this topic.
>the main difference between concerning ourselves with whether or not mathematical objects exists vs music
There is no 'vs', because there is no difference. Music is math, any song or sound is a mathematical object. A physical waveform that you hear can be encoded digitally in numbers: ones and zeros. So any given wav/flac file is just a bunch of numbers that give rise to the qualitative experience of sound, when interpreted in a certain way. For example, a digital waveform consists of samples, each sample takes 16 bits to encode. Sampling rate of 44.1kHz is 44,100 samples per second. So you have 16 bits per sample x 44100 samples per second per channel x 2 channels x 300 seconds = 2^423,360,000 possible permutations of a 5-minute audio file without compression, which is a number with over 127 million digits. A little percentage of these permutations would count as music (even if your tastes are really diversified), most of it would just be noise. But all these possible 5-minute audio files include not only every song and every performance that existed or will exist. They also include every possible sound recording: songs that will not be written, Paul Graham saying that he hates HN, Paul Graham saying that he loves me and the rest of the file is silence, you and me discussing this topic with Plato for 5 minutes, etc, etc. The data exists and can be discovered and listened to, even though some of these examples are obviously not physically possible (i.e. Plato is dead).
So all music already exists mathematically, and it can be a useful mindset that your job is to discover it. A lot of musicians see it that way, Tessa Violet, for example: https://youtu.be/QzBoGVToWEo?t=342
https://en.wikipedia.org/wiki/Harmonic_function
vs. this:
http://openmusictheory.com/harmonicFunctions.html
Maybe, at some fundamental level, they are the same...
But you cannot just "invent" a new proof to the Pythagorean theorem. Any proof that a^2 + b^2 = c^2 will have to clear the hurdle of being demonstrably "true."
Why not? Religious people believe that God is the Most Wise. Mathematics in nature attest to that attribute of God (as well as to other attributes of Him). An intelligent and religious person would recognize that, knowing it is God who came up with all the rules that keep the universe in balance.
I'll never understand why some people believe science and creationism can't go hand in hand.
I probably butchered this but it gives you some starting points to research and hopefully shows a model of the debate. Huge topic. Check out https://plato.stanford.edu/entries/epistemology/ as a hardcore crashcourse or check "crash course philosophy" on youtube for the relavent sections at a more HS level. SEP has entries on god, faith and mysticism as well as rationalism and empericism too.
I choose my words carefully. I get what you're saying though, but I think that's more of a subjective matter.
Of course this is hard to understand, but what would you expect? We can't even comprehend the tiny amount of His wisdom He has given us, let alone comprehend all of it.
If you're in the sahara desert you basically have all materials you need to create an iPhone. Not in a million billion years will there ever come an iPhone into existence by mere accident of natural forces in that desert, would there? Yet you believe that living things, which are not even comparable in complexity to an iPhone, were formed by a chain of coincidental events in the universe?
You can see God if you are willing to, unfortunately most atheists keep their hearts closed...
> You ascribe beliefs to me via stereotype.
Not that I feel offended by the first quote, since I don't believe in a hypothetical god, but how am I the one stereotyping you? If you don't believe in God, which you made clear in your first comment, then you disbelieve in Him. So that means you're an atheist, right?
Or am I misunderstanding the concept of being an atheist? To each their own of course.
The evidence says that actually did happen. Abiogenesis and the evolution of human beings were just part of the process of producing an iPhone through a "mere accident" of natural forces over the course of the last 4.5 billion years or so. Producing an iPhone was never the goal, of course–natural processes don't have goals. But it was one of the consequences.
If God can exist without being created, so can what you refer to as "His creation". This idea that the visible universe requires a Creator, who in turn does not require a Creator, is arbitrary and capricious.
The magic part for me is that some axioms have been chosen so well that their conclusions are confirmed in the real world.
Or in other words: the constraints on maths are imposed from outside of maths.
How do we know that it is true?
Sound almost like "jump off the roof and see what happens."
How would you revise this statement if we lived in a "Mathematical Universe", like Max Tegmark's hypothesis.
> The magic part for me is that some axioms have been chosen so well that their conclusions are confirmed in the real world.
It's actually hard to avoid Turing completeness, and once you have that, any recursively enumerable function is calculable. All you need is addition and multiplication on numbers.
https://youtu.be/orMtwOz6Db0?t=3777
Is that multiverse theory disproven by Wolfram's latest blog post?
"could there be other universes? The answer in our setup is basically no."
https://writings.stephenwolfram.com/2020/04/finally-we-may-h...
Just to take a recent example which was mentioned here, Geometric Algebra[1]. There you assume you have some objects which aren't numbers but which when squared equals a given number. By doing that a bunch of nice results have been discovered.
However to me the basic premise, take some objects which aren't numbers but which square to a number, feels very much like an invention. So as such wouldn't the nice results be inventions as well?
[1]: https://bivector.net/doc.html
Not really, because you have no choice about B - it's true (or rather it follows from A) whether you want it to or not. You could have picked a different A, but once you picked A then B was fixed.
At least if there is to be any meaningful distinction between a discovery and an invention.
The world of ZFC doesn't feel very constrained by the actual universe, is all.
And if you don't like ZFC, you can pick another set of axioms entirely- there are several alternatives.
Math is purely abstract science yet it describes the world around us to the utmost precision. Does that mean that this world is simply ... a math model which means everything around us is ... not real?
I've long stopped believing in free will because everything points at it being an illusion of our brain because we're a product of this world and we had no chance of influencing the conditions which brought us to life, and even after our minds and consciousnesses form it's hard to believe they are fully autonomous and not simply a function of the processess in our brains we're simply not aware of.
If you think about all of it, it becomes utterly depressing as you begin to realize you're a biological robot, a byproduct of the universe evolution which couldn't care less about our species and this little tiny blue planet.
You can only reach this conclusion if you categorize math as "not real". If everything we interact with is a perfect mathematical object, I prefer to take that as evidence that all mathematical objects may be real, however abstract they seem.
So no, our world is not unreal because it can be described with maths. Maths have evolved to describe it. Same as human language has evolved to support our close environment and our human interactions.
It does no such thing. The models we create using mathematical language do.
Math is a precise language that can be used to describe the world around us precisely. It can also be used to describe utter nonsense or utter fantasy precisely.
There have been a lot of efforts of modelling social phenomena, the arts, etc. mathematically and while there are some interesting partial results (e.g. that languages can to some reasonable extent be analysed by parse trees or some aspects of music), most "grander" theories I have seen do not really stand up to much scrutiny.
I'm not an expert, but I could assume that part of this is due to a lot of nonlinearity in the phenomena studies, which means that many classical methods don't work well; maybe chaos theory etc. could shed more light on these things, but I don't know enough about it.
Except it's not ineffective at all in these subjects. Social science experiments have so many variables that the small experiments that can be conducted given the financial resources are insufficient to infer a good model. This is not a math problem, it's a money problem.
But even if you could design an experiment perfectly and come up with some strong statistical evidence and then had other means to tease out what is actually causal and what is only correlational, you'd only know what influences what and not necessarily why. But yes, I did say that statistics/probability is some rare exception.
You can study group theory and understand the way physical forces work better, or hilbert spaces to understand quantum mechanics, but I haven't yet heard of anyone who has studied topology or galois theory and found that incredibly useful for understanding social phenomena better.
Except you have no way to conclude that it's ineffective. Analytical solutions require data to study. Without data, or with little data, what analysis are you going to perform? At best, broad statistical correlations, which is exactly what we find.
You're effectively claiming that spoons are ineffective at a restaurant that provides only forks. Well no, if a spoon were available, then it would probably work just fine.
> but I haven't yet heard of anyone who has studied topology or galois theory and found that incredibly useful for understanding social phenomena better.
After 5 minutes of Googling:
* Power laws: https://en.wikipedia.org/wiki/Power_law#General_science
* Network theory: https://en.wikipedia.org/wiki/Network_theory
* There's a journal specifically for mathematical social sciences: https://www.journals.elsevier.com/mathematical-social-scienc...
* Economics, game theory, and social choice theory are all examples employing heavy analytical problem solving to social problems
* Quantum mechanics applied to social sciences: https://www.cambridge.org/core/books/quantum-social-science/...
The main stumbling block is that mathematicians are interested in mathematical problems, and so they make a common but mistaken assumption that social sciences either don't have such problems, or they are too messy for elegant math. Take it from a mathematician, this is incorrect: https://www.mathtube.org/sites/default/files/lecture-notes/S...
When it comes to economics, I know that a lot of people don't agree with the basis of many mathematical models that are used, but I'm not an economist, so I can't speak to that.
So maybe I was overzealous in discounting mathematics for the social sciences altogether. Still, I would contend (albeit with much less evidence):
- There's some measure of people trying to construe "nice models" of things in those subjects instead of trying to make sure they agree with reality. I have a degree in linguistics and I've seen this over and over. Most of these models haven't convinced me at all.
- The amount of maths that you either need or at least benefit from in order to do good research in such areas is still substantially lower than in, say, physics. I think it's still to be noted how we can describe much of physics just with a small number of economic and elegant models. I haven't seen anything comparable in any of the social sciences.
It can also be liberating and uplifting. You're a part of the universe - you are the universe expressing itself. The result of uncountable generations of brutal evolution, you're quite a miracle!
Free will doesn't depend on the universe being deterministic or not. Things that haven't happened yet are likely going to happen in a certain way based on the trajectory, but it doesn't mean that you can't change your own path in a meaningful way, you're free to take a harder path vs. an easier or more obvious one.
This is of course debatable, there is some evidence to suggest that the feeling that you took a certain path is illusionary. Your brain made the decision, then you became cognizant of the options and you felt yourself making the decision[0], but if you could rewind the universe, you would always choose the same path based on your 'brain-state' at that time. At least that is what I understand of the most extreme "free-will doesn't exist" position.
Just got lost in the SEP rabbithole [1], still not sure where I land on this issue.
[0] : like how you feel thoughts 'bubble up' during meditation, you didn't actively 'think' those, they appeared to you. EDIT: atleast that is what it feels like
[1]: https://plato.stanford.edu/entries/freewill/#DoWeHaveFreeWil...
https://en.wikipedia.org/wiki/Our_Mathematical_Universe
(Obviously I'm in the religion where math does exist.)