Math is merely a mutual agreement. First an agreement on root principles, and then an agreement on notation for these root principles. The rest is built on top of those. Nobody is forced to accept the premises. (Unless you don't want an F in math class.)
One imagines two intelligent alien races would be able to communicate that pi is the ratio of the circumference to the diameter of a circle without coming to any real agreements over postulates or axioms. But I suppose that's natural philosophy, not math.
See https://computingbiology.github.io/docs/hamming1998.pdf "Mathematics on a Distant Planet" for more thinking. Many physicists think that math is instrinsic and absolute (objective) in the universe, not "merely a mutual agreement"
> One imagines two intelligent alien races would be able to communicate that pi is the ratio of the circumference to the diameter of a circle without coming to any real agreements over postulates or axioms.
Except circles don’t exist in the real world. They are an abstraction that mankind created that are similar enough to many natural objects as to be a useful abstraction of them.
In order for two alien races to communicate about pi, they’d have to first agree on the definition of a circle as a set of points equidistant from a central point.
That’s one of the axioms they’d have to agree on before they could even get to pi.
Assuming that the two alien races are indeed "intelligent" (approximately relative to humans), and that they possess sensory perceptors approximately similar to those of humans, and that they inhabit the same observable universe as humans do - a universe that contains a great many physical (imperfect) circles - surely they should be expected to have conceived of the same abstraction of a (perfect) circle? And therefore to have gotten to pi?
Then again, maybe a circle (and all other geometric abstractions) are too tied specifically to visual perception. Maybe, for example, intelligent aliens who only have auditory perception, would have no concept of a circle but would have gotten to the Doppler effect. Or intelligent aliens who only have tactile perception, would have no concept of a circle but would have gotten to the laws of thermodynamics.
How do you agree on those root principles? Don't you need a set of even lower level shared rules and ability that allows you to agree on anything in the first place? If I show the paper with the rules to a dog, I don't think they can agree or disagree with me in any meaningful way, despite otherwise being reasonably intelligent creatures. Could there be aliens that are vastly different from us that can make rules that we would be unable to agree to? Could there be aliens similar to us, that would have different means to agree that are incompatible with ours?
Mathematics is, in part, the study of universal traits:
(1) Ratios between things in the world.
(2) Logical relations between things in the world.
(3) Absolute distinctions between things in the world, in a nominative sense, which is to say in the sense that numbers can be assigned to things or elements of things.
When mathematics does not relate to one of the three -- for e.g., in Cantor's theories of "countable" and "uncountable" infinite sets -- it is totally unreal. A construct or game played with logic that has no prior or intrinsic relevance to the material world.
Math is in language. In a similar way that we can have fiction and non-fiction, with different uses of language, you can have 'pure' and 'applied' uses of math. Using 'real' as the goal kind of misses the point. It's not imaginative, if that's the antithesis of real in this whole conceptual discussion. But, like fiction to non-fiction, it's still a public phenomenon that "touches" the world in language as a shared, communal, element in a form of life. Math is remarkably useful as a representation of the world and for us solving problems, but you can also give directions to a location to someone with just hand signals and barking like a dog. In a way, math is not special. But, the degree of accuracy that we've developed in math, however, is what makes it remarkable.
> Using 'real' as the goal kind of misses the point
It misses the point of the question "is math real"?
Tbf, i am actually sympathetic to the position that "realness" is not really relavent or a well defined term when it comes to what is basically a descriptive language of patterns and relations. But "is it real" is the question we started the thread on.
The study of math in language is called semiotic, and thanks to a bunch of french poseurs it’s become a joke word among the genuinely intellectually curious. Nevertheless, attempts have been made to tie it all together, most notably by the brilliant and mostly forgotten Charles S. Peirce.
Whence the pragmatic maxim:
Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object.
Pragmatically speaking, math is no more or less real than that.
Okay, here is precisely where the Simulation Hypothesis becomes useful and interesting. I know, I know. But bear with me for a minute...
The universe, as we know it, is simulatable. In other words, it can apparently be reduced to mathematics in precisely those three senses I outlined above. (If it helps, you can imagine electrons and other subatomic particles as bits of information in a coordinate space, which are constrained to operate in accordance with rules that govern logical relations between things.)
What mathematics is real? Anything that would relate to that universe or any of its constituent parts, in _any_ meaningful sense. We don't need to have found a use for it, and surely there's a great deal that is still undiscovered.
What mathematics is unreal? Whatever is, a priori, absolutely unnecessary to the existence of such a universe -- or, even worse, would break a universe described in terms of logic were it somehow made manifest.
Given that logics (of all sorts) are subsets of mathematics, I'm not sure one can convincingly hoist (any) logic above all other mathematics to make arguments.
> When mathematics does not relate to one of the three -- for e.g., in Cantor's theories of "countable" and "uncountable" infinite sets -- it is totally unreal.
Are you considering the set of natural numbers to be “unreal”? Or, if you consider the set of natural numbers to be real, do you not consider power sets to be a logical relation?
To be fair it's not too strange to consider both the natural numbers and the power sets to be to greater or lesser extents abstractions to allow us to reason about all the things we can logically put in one of them.
After all both of them are mostly full of things we will never need, encounter, or even be able to define.
I'm not a finitist, but it's a highly defensible philosophical position.
As for this:
> Or, if you consider the set of natural numbers to be real, do you not consider power sets to be a logical relation?
Skolem solved that quite neatly already: Every "uncountable" set has a countable model. Thus the power set is, in fact, no larger than the set of natural numbers, because both can be fully described in a countable manner. What I'm describing is necessarily an abstraction of an abstraction, though, so I'd consider it "unreal" by definition.
I know what “a model of a theory” (such as a model of a theory of sets, even one which asserts that some expression defines an uncountable set) means.
I’m not quite sure what a “model of a set” means.
I suppose maybe you mean like, the set in some model, which is described by the given description of a set in the theory?
> Thus the power set is, in fact, no larger than the set of natural numbers, because both can be fully described in a countable manner.
I don’t think this follows.
There are countable models of set theory. And for these models, there is (in the meta-theory) a bijection between the set representing the set of real numbers, and the set (in the meta theory) of natural numbers.
I don’t think this establishes that “in fact” there is a bijection between the real numbers and the natural numbers.
Rather, in any model of any of the usual set theories, there will be no bijection between the set of real numbers and the set of natural numbers. (Of course there will not be, because these theories entail that there is no such bijection.)
If I take a non-standard model of arithmetic, and for some non-standard natural number n, and consider the uniform distribution of (non-standard) natural numbers less than n, then for every standard natural number, the probability of getting that natural number from that distribution, will be equal. Are we therefore to conclude that there is a uniform probability distribution over all the standard natural numbers? By no means!
Skolem's own proofs are a little bit opaque, so I've constructed a new algorithmic proof of the downward Löwenheim-Skolem Theorem which is easier to comprehend. I have yet to publish it, but it'll probably turn up somewhere else towards the end of the year:
Preliminaries: (1) Let T be a first-order theory with a countable or uncountable infinite model M. (2) Let L be the language of T. (3) T is assumed to be a set of sentences (closed formulas) in L.
Algorithm Steps:
Initialize Countable Set S: Start with S = Const(L), the set of all constant symbols in L.
Extend Language: For each n-ary relation R in L and each n-tuple (c1, ..., cn) of elements in S, if M⊨R(c1, ..., cn), then add a new constant symbol c to S and extend L to L' by adding c.
Iterate for All Formulas: For each formula ϕ(x1, ..., xn) in L and each n-tuple (c1, ..., cn) of elements in S: If M⊨∃xϕ(c1, ..., cn), then pick some a in M such that M⊨ϕ(a, c1, ..., cn). Add a new constant symbol c to S to represent a and extend L to L' by adding c.
Closure: Repeat Steps 2 and 3 until S no longer changes. Since T and L are countable, this process will eventually result in a countable set S.
Construct Countable Model N: Take the substructure of M generated by S as N. By construction, N is countable.
Elementary Submodel Check: By construction, N is an elementary submodel of M. This is because for any formula ϕ and any n-tuple (c1, ..., cn) from S, M⊨ϕ(c1, ..., cn) if and only if N⊨ϕ(c1, ..., cn).
Conclusion: N is a countable elementary submodel of M, and therefore T has a countable model. End.
You're right about the bijection, but, in the countable model, the set that "represents" the real numbers is countable. (From the perspective of the meta-theory.) However, within the model itself, this set still satisfies all the axioms that make it "look" uncountable. For example, there's no bijection between this set and the set representing the natural numbers within the model, even though such a bijection exists in the meta-theory.
This is why Skolem's solution can be most succinctly described: "Every set has a countable model if one steps outside the set and constructs a countable model from its elements."
In Kleene's “Introduction to Metamathematics,” he describes the situation as follows: "Either we must maintain that the concepts of an arbitrary subset of a given set, and of a non-enumerable set, are a priori concepts which elude characterization by any finite or enumerably infinite system of elementary axioms; or else (if we stick to what can be explicitly characterized by elementary axioms, as we may well wish to in consequence of the set-theoretic paradoxes ) we must accept the set-theoretic concepts, in particular that of non-enumerability, as being relative, so that a set which is non-enumerable in a given axiomatization n may become enumerable in another, and no absolute non-enumerability exists."
I was going to post something similar and then saw yours, and it was better. We can add that math is language, and sometimes language describes things that are real, and sometimes not.
> When mathematics does not relate to one of the three -- for e.g., in Cantor's theories of "countable" and "uncountable" infinite sets -- it is totally unreal.
I find that to be quite a bold statement. The question as to whether "mathematics is real" feels firmly rooted in philosophy. First, I would question what is meant by real and unreal. Second, I would question if the answer matters whether the existence of the real and the unreal are not yet discovered, or even imagined.
Your describing physics more than math in my opinion. But it's hard to say things like set theory isn't real when a lot of calculus works nicely and describes the world very accurately. That toolbox is heavily built on the "unreal" things in math your describing to be not intrinsically relevant. The weird parts of math often speak to some deeper thing your discounting. Hopefully I'm not reading into your comment to much
I think the tie between Mathematics and the real world is more of a fortunate coincidence than a necessary tie or even causality. Human will still study math (as we know it now) if it real world doesn't exist or it somehow becomes not describable by math, because there's the innate desire for logic and solving puzzles.
There is, almost certainly, an objective universe that exists outside of human minds, and it also seems likely that all of mathematics can be determined through objective mechanisms.
This is an idea that goes back to Pythagoras and Plato.
Think of a triangle. Now draw that triangle on paper. If you look closely enough, you'll see "imperfections" in the triangle you just drew. Now ask yourself: "how do I know this thing I just drew is imperfect? Where did the idea of a perfect triangle come from?"
Plato would say the perfect triangle comes from the realm of "forms". This mystical place which is "more real" than "reality" because everything there is perfect and everything here is just a flawed approximation. Plato also said that this is the place where our souls go when we die, and we engage in "congress" with the forms and then return to earth, reincarnated. When we learn things, we aren't learning something new but actually recalling memories of the forms. This is why everyone knows what a perfect triangle is but no one has ever seen one in the physical world.
Because the Platonic soul (psuche) is different from consciousness. Actually, Philolaus (first Pythagorean to write a book) said that conscious feelings come from the combination of the mathematical soul with the body. So, Pythagorean soul can be viewed as the set of logical or conceptual forms— which constitute a person, yet can also be passed on from person to person. That’s a very different kind of reincarnation…!
As a data point, the author was interviewed on NPR's "The Indicator"[1], and the way this review summarizes it might not capture what her point was:
> CHENG: I'm not trying to answer whether math is real or not. I'm trying to show that considering the question at all leads us to interesting thoughts. And in the end, what I say is that, with all these questions, I don't think there are yes-or-no answers, and we shouldn't claim that there are. What we should do instead is say there is a sense in which - you know, what is the sense in which math is real, and what is the sense in which math isn't real?
> And the thing is, I think a lot of people who say math isn't real are using that to say, oh, so it's irrelevant and stupid. Why should we study it? It's made up. And what I want to say is that just because it is made up doesn't mean that it's irrelevant. And actually, the fact that it's kind of made up makes it really powerful because - well, it makes it really, in a way, more accessible because you don't need a lot of money to get it. All you need is an imagination. And I think that's a really amazing thing about it. And just like fiction isn't real, but fiction can give us insights about the world around us to highlight much more, specifically, things about society. And that's what I think is powerful about abstract math as well - because we're not constrained by reality.
I haven't read the book, but it seems like the title may be a throwaway question meant to pique the reader's interest and say "let's get philosophical about math".
You might as well ask "Are ideas real?" in the broader philosophic sense. Yes, they (with math as a subset) are real existents within consciousness. The question of mathematical ideas' connection to reality is epistemological. "Real reality" pertains to metaphysics - the externally perceivable stuff of reality that exists independent of our minds (as in, eternally prior to, and subsequent to, the existence of any humans or other beings with conceptual consciousness.) Math is an epistemological tool to abstract the causality of existence into a simplified structure amenable to consideration and manipulation by human consciousness.
A quadratic equation can be used to approximate the coordinates of the trajectory of a thrown object in a gravity field. I think the really interesting question is why that particular equation reflects that trajectory. Arithmetic operators - multiplication and addition in that case - in a particular order, are approximating the causal operations of existence that are actually at work. I think of this as the philosophy of mathematics and much more should be done to investigate it.
> Math is an epistemological tool to abstract the causality of existence into a simplified structure
The mathematics to accurately predict or relay reality is still complex enough that it’s often beyond us. You’re right in that it’s a tool to understand, but if we’re using simplified math for simplified reality, is it really epistemological?
As you suggest, math isn’t outside the boundary of philosophic investigation. It never was in the past, and I don’t think better approximations change that calculation.
It's perceivable as long as a consciousness exists with sense organs. Existence doesn't depend on that perception. Existence existed long before any minds existed and will it exist long after. ("Long" is a euphemism here for eternally.)
"Within consciousness" doesn't belong here. Even if all sentient life in the universe died out tomorrow, the idea of triangles (and "triangluarity") would still exist.
well… on the other hand: have been triangles ever experienced without a consciousness?
this "the idea of triangles would still exist" statement looks very reasonable, but can we ever proove it? you can not remove ALL consciousness to verify. at least a bit consciousness must be in the system to do the experiment.
yes, it very much coincides with all of our understanding of reality, that abstract ideas like triangluarity is independent from any consciousness. at least any consciousness known or conceivable to us so far.
so my extension to this statement would be - admittedly supernaturally-sounding - that: Even if all sentient life in the universe died out tomorrow, the idea of triangles (and "triangluarity") would still exist, possibly because there is a consciousness above/beyond of the conceivable the universe to sustain their existence.
We're very quickly coming to the realization that information is a physical quantity, like mass or energy. (And can be measured and follows some sort of physical laws.)
If so, then consciousness doesn't need to exist for information structures to exist.
Brains model patterns in signals, math could be described as the brain's language for expressing higher order manipulations on all those models with communicable symbols. So it's kind of both real and invented, because brains invent, and brains are also just real objects and any models they form came from somewhere.
Perhaps it's acceptable in moderation, like "ain't". It's needed for a rhyme in a well-known song by Billy Joel, for what it's worth. On the other hand, there is a different song by Billy Joel in which "anyway" is used for a rhyme.
No, I'm not a Billy Joel expert or fanatic. I just happened to notice, all right?
The notion of existence is doing a lot of the leg work. A epistemological solipsist can say things "exist" when they have a model of their subjective experience in which things can exist. It gets real funky because your interpretation of qualia results in a model that includes the brain and thus a model of how you are interpreting things as a model in your model.
> Oftentimes, math is taught as a set of rules and processes divined from an authority in which learners must memorize the rules and strictly follow the processes in order to satisfactorily find the one and only correct answer
I was surprised later in life when the majority of people I talked to felt Math was this way (and a good thing). To them, math is a set of rules you learn to follow to the T and use them on other problems.
For me, everything a Math teacher conveyed was more of a recommendation, a suggested tool that I could incorporate into my tool box. The methods they employed to solve problems were a matter of preference to me, rather than rigid rules.
This way of thinking has always had some pros and cons. I never solved a problem the way a 'grader' was expecting. Some teachers loved the creativity and efficiency, other TA's just marked as zero. It made applying what I learned to other things, but on exams I would always be stressed with time because I would spend time on which way I was going to answer the problem.
That being said, I am very good at Math. scored well in HS/college, 165/170 Math GRE score, in a field where Math is important (Data Scientist).
And in addition a real number can encode an infinite amount of information, and as such real numbers are a construct that doesn't exist in the physical universe. Making them, unreal.
This book is on my reading list. I listened to an interview of the author who said that their primary motivation of the book was less about teaching math, but teaching the value of looking at the world abstractly to determine patterns and constructs to explain our shared experience.
Is math real or not? It doesn't matter she posits, it works and continues to work and we can learn from its existence that almost everything can be explained and is "predictable" given enough inputs.
I've encountered this argument before, but also applied to other subject such as eg Latin:
> 3) math has an indirect usefulness which is a way of thinking that is transferable to a myriad of disciplines and solutions to problems in everyday life. And it is this third reason that makes math relevant for most people.
Are there actually any good studies that show this in a counterfactual setting? Like, do we actually know that spending time teaching maths (or, less plausibly, Latin) helps students aquire these abstract skills more than other subjects?
Is this "transferability" of meta-skills a testable outcome?
I’m skeptical of (3) as well, and yet I encourage students to study math through calculus, and to study Latin.
My feeling is that the utility of higher math, Latin, history and literature comes every minute of every day, as you experienced life as someone who has familiarity those things and your life will be richer and fuller.
This is decidedly not testable. And yet I still believe it.
That (3) is a very interesting issue. My suspicion is, it basically says "knowing logics is good for your well-being".
The skill of logical inference — even at the level of very basic syllogisms — is both very much underappreciated and underdeveloped in the American college population, at least from my personal experience. As good citizens, we all collectively should grab a couple of Martin Gardner's or Lewis Carroll's books off the shelf and give them a good read. I predict it will do much good... and if I'm wrong, it certainly won't do any harm!
There are plenty of ways to use this stuff in everyday life.
I took logic in college, and although the "logic as english statements" stuff was sort of confusing, the symbolic stuff like A&B = !A|!B stuff has helped with computers all my life.
It was only much later in life that I ran back into logic as english statements in a way that made sense as practical.
I read a book where they took apart the statement:
If you loved me, you would take me to the movies.
Most people in relationships will respond to this with:
Well, I just took you to the movies last week! Why do you want to go again tonight, we had other plans! etc...
But the book explained that with "If X, then Y" it was futile to address Y. You must address X:
Wait, do you think I don't love you? Of COURSE I love you!
...just hard to unpack this in the middle of an emotional situation unless you've studied the logic
An intro to formal logic class was the first thing that made me think of becoming a professional programmer. The class had a lab portion that used a program called Tarski's World that I remember as being a lot of fun.
I would highly advise against treating informal natural language statements as logical statements. It is impossible to know what someone who says "If you loved me, you would take me to the movies" actually means without far more context. The fact that it has the form of an X=>Y statement is at best a hint, but it definitely shouldn't be taken as literally as that.
I feel I became much better at writing non-fiction after taking math at university (mainly calculus, linear algebra and statistics).
Going through proofs and proving things on your own really transferred to being able to better present arguments. The diversity of the math I learned has helped to reflect on things from different perspectives.
In sum this helped with everything from thinking more and better about the core issue at hand, writing argument chains in the correct order, cutting down on irrelevant stuff and more.
I've used this to significantly help the grades of both my SO and a family member, who both took non-math topics, by improving their hand-ins. I didn't know their field so was strictly improving the structure and presentation, and asking for clarifications where I felt the arguments didn't add up, and have them write down the answer.
I feel it still helps me a lot writing emails at work and similar.
I actually think point (3) is a good one, but am still extremely skeptical of teaching students more maths.
Basically, I believe that there's a heavy correlation between being good at maths and being good at solving every day (and not so every day) problems.
But I don't believe there's much of a correlation between being taught math at school for even more hours will make much of a difference. Most schools are terribly at teaching anything.
And, since LLMs are so bad at math currently, we may find that, by improving their math ability with gobs of synthetic data, we get improvements in general reasoning.
Thanks, this is a nice reference!
Interesting that the abstract and introduction mention the lack of existing evidence - seems like an under-studied question?
One aspect that's quite easy to criticise about this study is that it uses existing groups of students with different levels of maths training. This means that there is possibly self-selection etc, and one may argue there might also be a causal effect in the opposite direction (e.g. folks that are good at reasoning like to do maths.)
> Are there actually any good studies that show this in a counterfactual setting? Like, do we actually know that spending time teaching maths (or, less plausibly, Latin) helps students aquire these abstract skills more than other subjects? Is this "transferability" of meta-skills a testable outcome?
We know that if you try to train someone in math (or in Latin), and they do well, then they will also do well at other things in the rest of their life. Some people would like to give the credit for that good performance to the Latin training.
Well, what do you mean when you say something that "exists physically" is "mathematical"? Depending on your answer, the answer to your question could very well be "everything".
For example, we could ask whether anything that we think of as "existing" would be missed by simulating the equations of physics with the empirical starting conditions of our universe.
There are real limits in what humans know about the universe. But it's possible in theory that everything that is "real" follows from some differential equations.
In theory, maybe. I don't think our models of the universe are anything other than imperfect but useful approximations though.
Could your question be rephrased as "take the real world, with its things we don't have an accurate model of (which is most of them). Would we notice if we replaced this universe by a tidier one, perfectly described by a math model?".
It's an interesting, scifi question alright. I don't know if it has an answer.
Alternatively, if mathematics cannot, yet, completely describe reality, it is only because our understanding of mathematics is an imperfect approximation of 'true' mathematics. As our understanding of mathematics improve our descriptions and models of reality will improve asymptotically.
First, what would our understanding of mathematics improving look like? Mathematics is an abstraction. Does such a thing as "true" mathematics even exist?
Second, what evidence is there that improving our understanding of maths will asymptotically improve our models of reality? There really is no evidence of this! Seems like a wild logical leap.
This is a curious statement. Mathematics is a description, not a prescription. When Newton says that the path of a projectile is parabolic, he does not mean that it is in the shape of a parabola. He means that a parabola is a very good way to approximate the shape of the motion.
> Mathematics is a description, not a prescription.
But are you sure about this? If mathematics were not prescriptive then we should expect to see some evidence of that. So far nobody has demonstrated the existence of phenomena that contradicts the equations of physics. Usually we would call such phenomena "supernatural."
It's possible in theory that nature doesn't have to obey mathematical laws. But if there's no evidence for it and if it doesn't explain any phenomena, then what is the argument for holding that particular belief?
Physics doesn’t obey mathematical laws. Mankind studied physics until we found mathematical formulas that describe nature reliably. There are countless math formulas that were discarded over the millennia after they were discovered to not adequately describe the physical phenomena they were intended to describe.
Math is a collection of abstractions. Some are useful in explaining natural phenomena. Some are not.
The most straightforward evidence that mathematics is descriptive and not prescriptive is Einstein's General Relativity which illuminated that Newtonian mechanics was not prescriptive. Similarly General Relativity does not prescribe what happens at quantum scales.
> So far nobody has demonstrated the existence of phenomena that contradicts the equations of physics. Usually we would call such phenomena "supernatural."
You cannot contradict a description; it's not a "law" in the legal/prescriptive sense.
How would you even "contradict the equations of physics"? The real world exists and it's there; we use models to imperfectly describe (to varying degrees of success) its phenomena. In a sense, the real world contradicts those equations because they do not perfectly describe it; they are just an approximation.
> So far nobody has demonstrated the existence of phenomena that contradicts the equations of physics.
That's kind of a tautology because physics is rules for what we observe. If we observe something contradicting the rules then we come up with new rules to describe the new thing.
Curiously, this review (and perhaps the book being reviewed?) makes no mention of the model of mathematics as a language. This seems to be a common idea among working mathematicians and physicists[1,2], and it is rather clear that the question of reality of a language is ill-formed. A rich enough language should be suitable for expressing both real and abstract statements. It is also interesting to think about mathematics from the point of view of programming language paradigms: is it functional (I think mostly), what is its type system, etc.
Mathematics is the language best suited to communicate Physics.
> The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed.
F is a concept of force and has a wide variety of English words that apply.
Mass is represented in English by the tokens: "the property of matter resistant to acceleration"
Acceleration is represented in English as the change of the change of the position through time.
To your point, there are an infinite number of definitions we can apply to the equation defining a product. So Physics context + mathematical verbiage = elucidated ideas.
You can disagree with him that it's the best language or that he didn't prove his case, but I don't see how he's offering a strawman.
I'm also not sure that translating without context is a meaningful. What language operates without context? Without context, what would the pronoun 'that' in your post point to? What does the phrase 'back to' mean without the associated physical metaphor? How would this be different from translating to and from German without context?
> Curiously, this review (and perhaps the book being reviewed?) makes no mention of the model of mathematics as a language. This seems to be a common idea among working mathematicians and physicists[1,2], and it is rather clear that the question of reality of a language is ill-formed.
Sorry, if this isn't clear to everyone, why does this result in the rejection of mathematics as a language rather than the rejection of the question of the realism of mathematics? The latter seems much easier with no downsides (that I can see), whereas the rejection of mathematics as a language has many downsides.
Of course, I do think of math as a (meta)language. Maybe my phrasing does not convey that well. So much for using natural language instead of math... ;)
They probably didn’t mention it because, like you pointed out in the sentence immediately after your question, it makes the philosophy ill-formed and dare I say uninteresting.
Are languages real? Sure, I guess. Why not. They’re made of concrete sounds and symbols that everyone (by design) can point to - they might even be the realest thing there is.
Are (math) concepts real?
As in like, if I rearranged all the matter that exists out there, would the value of root(2) stop being irrational?
Or maybe they don’t have to be physical to be real; maybe being “real” has more to do whether or not something has an effect on something else than its physical existence. Then, there’s a difference between how something exists and how real it is?
Etc.
Thinking of math as a language as opposed to a set of concepts isn’t interesting to the discussion of its reality IMO
1. I do not think Galileo is either a characteristic example of a physicist or mathematician, or a particularly good example at all [1].
2. While I would say that most would trivially agree that mathematics is a form of language [2], it is less common to argue that mathematics is _just that_. And, crucially, it is not supported by neuroscience [3]. It seems that there are distinct brain areas involved specifically in mathematical thinking outside of those that are language-specific, which would mean that mathematical thinking is more than language [3].
[2] Actually, a collection of languages, as different parts of mathematics form different languages or dialects that are not easy to communicate in between even if they refer to similar objects, eg category theory vs set theory, analysis vs probability theory.
> And, crucially, it is not supported by neuroscience [3]. It seems that there are distinct brain areas involved specifically in mathematical thinking outside of those that are language-specific, which would mean that mathematical thinking is more than language [3].
I would not construe a small study of 30 people (15 math experts) as having the backing of the field of neuroscience.
The steel man position of mathematics-as-language would be something along the lines of "Mathematics is mostly a language + some other stuff". The paper in question only shows there is "some other stuff"---but that is also true of almost every non-contrived use of language. The paper shows that the other stuff for sentences about history and nature appear differently in the brain than the other stuff for sentences about math, but that is far from demonstrating that math on the whole is not mostly a linguistic phenomenon.
Personally, I think case studies of famous mathematicians or physicists discussing their thinking process (e.g. Einstein) is more convincing evidence against the hypothesis than fMRI studies.
I am no philosopher, and only an amateur mathematician…
But to me it seems that even when you consider math as just being language… you still run into the same philosophical problem when you consider what the semantics of that language are.
For example, you may believe that the practice of algebra is simply operations on stings of a language. But when you start asking what objects those sentences refer to, you end up asking what math is again.
This is where I finally landed with this question - there’s too much vagueness and importantly no agreed definitions of what is and is not “reality”
Math has no fundamental non-differentiable epistemology qua mathematics.
As you said there is no meaningfully discrete, bounded “object” that can be used as a universal reference - so the fact of relativity means that even if maths were “real”
I’m not even sure such a thing exists (who says the rules of the universe are uniform, or static as we’re only have a few centuries of poor observation) and if it does exist, if humans have the ability to identify it
This is a good point, but natural language has this problem to an even greater degree. What object does the word "consciousness" refer to? "Unicorn"? What makes a ship a ship? What if we include toy ships and drawn ships?
"What can math describe" is an easier question than "what is reality?" or "what is the nature of all concepts a human mind can conceive of or be interested in?". Not that it's an easy question either.
But we don’t expect natural language to be as precise and consistent as Maths. You could argue that maths was invented because natural language was too vague.
I don’t think Einstein could have come up with general relativity if he had to rely on any other language than math. So I suspect math is not “just” another language.
Unfortunately, it takes extraordinary effort by the math teachers to make Mathematics interesting to students. Even if some of the teachers are willing to put the effort in teaching this wonderful subject in a way that makes sense to students, it's not possible to turn all mathematics teachers into teachers who understand how mathematics MUST be taught.
I hope all these books, YouTube videos, and websites can help in making students curious about mathematics and explore further on their own. It is hopeless to even expect the school systems in US and a lot of other countries, to change the way their school systems work - teachers are not given enough time to spend on the topics in mathematics. It takes lot of extra effort by the students.
The actual reality is most students don’t have the cognitive horsepower for much past long division. I’ve seen adults struggle with basic algebra: “but what is X?” And so on. And that’s fine. Billions of people are going to live satisfying and fulfilling lives without ever knowing the chain rule.
I think you inadvertently highlighted what I suspect is the real issue; a ruthless focus on the path to calculus to the detriment of other useful maths.
I think I disagree! Calculus is great and all (truly), but logic, boolean algebra, probability, statistics, even linear algebra all have really interesting and useful insights and are very approachable. I'm sure there are other maths I've never encountered that can also be enjoyed with minimal pre-requisite knowledge, though that might be more in the 'abstract and difficult' territory.
The bits you need from calculus to approach those other subjects is also very minimal and approachable relative to the content of a calculus pre-req (where applicable).
Because ℚ ⊂ ℝ of course, but it is also not that far off from the conclusion of the book in that "math is real because it is an idea and ideas are real". In a non-mathematical sense, rational means "based on reason", and reason is the power of the mind.
That seems like a very weird statement to make. Many numbers in the real world are integral (two objects, one electron, etc.). Thanks to quantum physics, most measurements are integral too.
Explained better above. There are more irrational numbers, almost guaranteeing any number come across in nature is irrational. Interesting thought since I think one thing that makes irrational numbers is there is no function for them. So it's kind of a cheeky way to say no math formulas can ever describe the real world since all the numbers are irrational.
There are more irrational numbers, almost guaranteeing any number come across in nature is irrational
There is nothing that says that the distribution between rational and irrational numbers that show up in nature is the same as the distribution in our construction of the real numbers.
You’re missing the point: physics has shown the world to be discretized. Almost every number you go out and measure IS ACTUALLY AN INTEGER. In that sense the real number system doesn’t exist. It’s super useful, yes, but it’s an abstraction away from reality.
I thought that "infinities of infinities" ala Cantor exist though. I agree with you that reality being quantized means everything in physical reality can be normalized to integers.
I am a physicist, and I believe what I said is accurate. Although yes I am (1) oversimplifying, and (2) assuming that some form of quantum gravity is correct.
To expand a bit with some examples: the energy a photon in a standing wave in a specific cavity can have is quantised. But a photon out in space can have any old energy it wants to. (Of course, a given energy level will correspond to a specific wave length etc.)
Similar, an electron in a single isolated atom has specific quantised energy levels. But if you look at the electrons in a hunk of copper, they are essentially free to absorb and emit energy in almost arbitrary amounts.
An even stronger example is time: as far as I am aware, time is not quantised in any of our accepted theories.
There's some reasonable speculation that ultimately everything is quantised at the Planck scale, including time. But that's just a very reasonable hunch, not something that 'physics has shown'. (And you already point out that trying to marry quantum mechanics with general relativity is a hot mess.)
I agree that 'quantisation all the way down' is the way to bet. But that's just speculation.
(But I strongly disagree with your claim that physics is build on integers. Yes, it might be discretised, but there are plenty of discrete structures that are not integers. Look at a Rubik's cube for a simple example. On top of that: almost any real world measurement is better described by a probability distribution than by single number, be that an integer or otherwise.)
It is better stated that: since there are "so many more" irrational numbers than rational ones, if you were to pick a real number "at random," the probability that it would be rational is zero. The "many more" and "random" ideas are made precise in measure theory (and elsewhere).
There is an infinite quantity of both rational and irrational numbers, so isn't it therefore impossible for there to be more of one than of the other? Or is the reasoning that, because there is an infinite quantity of irrational numbers between any two given rational numbers, there are therefore many more irrational numbers than rational numbers? I would have thought that there being an infinite quantity of both, makes it impossible to compare the quantities.
There is a mapping from counting numbers (1, 2, 3, ...) to rationals and back again that shows these quantities are the same; for every element in set A there's an element in set B and vice versa.
This is not the case for irrationals... therefore it is concluded that the infinity of irrationals is a larger infinity than the infinity of rationals.
> There is a mapping from counting numbers (1, 2, 3, ...) to rationals and back again that shows these quantities are the same; for every element in set A there's an element in set B and vice versa.
That is true, but it's never taught. I don't even know what that mapping is, though I've seen it mentioned once in a popular treatment.
What's taught is always the mapping from naturals to rationals that overcounts the rationals, hitting them all an infinite number of times. (Because it's very easy to show a bijection between the naturals and the ordered pairs, but while (2,3) and (4,6) are distinct ordered pairs, they do not represent distinct rationals.)
But then all you've shown is that the naturals are at least as numerous as the rationals. To show that the naturals and the rationals have the same cardinality, you either rely on the idea that the naturals are the smallest infinite set, or you appeal to the fact that the naturals are a subset of the rationals.
There are also infinitely many rationals between any two distinct irrationals.
My favorite way of visualizing the difference uses the fact that every rational has a repeating decimal after some nth decimal place, and no irrational has a repeating decimal. Say you want to construct a number x, where 0 < x < 1, by drawing integers 0 through 9 randomly from a hat. Each integer drawn from the hat is placed at the end of the decimal; for example, if you draw 1,3,7,4 then the decimal becomes 0.1374. You then draw, say, 1, and it becomes 0.13741, and so on. If you could draw infinitely many times from the hat, what is the probability that you'll construct a number with a repeating sequence? That would give a rational number.
> because there is an infinite quantity of irrational numbers between any two given rational numbers
Indeed, you've grasped the core of it. There's no rule you can write for irrational numbers such that "b is the next number after a", because there are infinitely many numbers between a and b that you'd be missing. You can't count them, i.e. you can't map them to integers.
While the thrust of your argument is correct, you're missing an important point. There are infinite number of rational numbers between any rational a and b as well, and the rational number don't have the concept of the 'next' number either. Yet the rationals are Countable.
The argument as to why the irrational numbers are uncountable and the rationals are countable is more involved than what you've made out. But very simply you can think of it as you need an infinite string of digits to describe each irrational number, but each rational number can be written as two finite strings of digits (in the form A/B, where A and B are integers). So to write our the irrationals you have an infinite number of strings, where each string is also infinitely long, while with the rationals you have an infinite number of strings, but each string is finite.
> the rational number don't have the concept of the 'next' number either. Yet the rationals are Countable.
That's literally the same thing. What is counting if it isn't being able to say what the next thing is? Do you have a mapping to integers or not? If so, then every n has n+1.
I know it was more complicated, but jaza had the essence of it. Without what they observed the whole thing falls apart. Yeah, it still needs proof, but I'm pretty sure five other comments went there.
> So to write our the irrationals you have an infinite number of strings, where each string is also infinitely long, while with the rationals you have an infinite number of strings, but each string is finite.
You've set the table but forgotten the feast! You're missing the step where you demonstrate that there's a number that isn't in this list. (Hint: think diagonally.)
What is counting if it isn't being able to say what the next thing is? Do you have a mapping to integers or not? If so, then every n has n+1.
The point I was trying to make is that there is no concept of 'next' inherent to the rationals, nor is there any natural or canonical ordering. The ordering and what comes 'next' is entirely a property of which arbitrary mapping you choose (I'm partial to Gödel numbering). The resultant order that your mapping imposes on the rationals is rarely useful or meaningful.
The rationals are a totally ordered set. There definitely is a natural, canonical ordering to the rationals. It's the same numeric-magnitude metric we use all the time. 1/3 is less than 2/3.
That ordering doesn't have the property that all sets of rationals contain a least element, or that any rational has a successor rational. (That would make them "well ordered".) But it's a natural ordering.
>> The argument as to why the irrational numbers are uncountable and the rationals are countable is more involved than what you've made out. But very simply you can think of it as you need an infinite string of digits to describe each irrational number, but each rational number can be written as two finite strings of digits (in the form A/B, where A and B are integers). So to write our the irrationals you have an infinite number of strings, where each string is also infinitely long, while with the rationals you have an infinite number of strings, but each string is finite.
This argument doesn't actually work. If there were only a countable number of irrational numbers, you could specify them all fully by doing no more than a countable amount of work, even stipulating that describing a single irrational number requires listing a countably infinite number of digits.
>> because there is an infinite quantity of irrational numbers between any two given rational numbers
> Indeed, you've grasped the core of it.
What? That's not the core of anything. It tells you that the irrationals are dense in the real number line. You know what other set is dense in the real line? The rationals.
How is that possible? We've made the same observation about the irrationals and the rationals. We want to make a followup observation that is true of the irrationals but not the rationals. Our first observation obviously can't be related.
> There is an infinite quantity of both rational and irrational numbers, so isn't it therefore impossible for there to be more of one than of the other?
Mathematicians can even meaningfully compare infinities.
You can also look at eg a uniform random variable on the interval between 0 to 1. The probability of hitting a rational number is 0%. The probability of hitting an irrational number is 100%.
> Or is the reasoning that, because there is an infinite quantity of irrational numbers between any two given rational numbers, there are therefore many more irrational numbers than rational numbers?
No, that's not enough. There are also an infinitely many rational numbers between any two given irrational numbers.
If you actually look at how real numbers are constructed. They are quite bizarre. The simple concept of the number line becomes a quite complicated set of sets that follow certain conditions.
(Sqrt(2) as a real number, is actually encoded as the set of all rationals less than sqrt(2) on the number line).
I've come to the conclusion that all numbers in the real world are integers and real numbers are a human construct necessary evey time we select a unit that is too large. There is evidence (e.g. quantization) that at the most fundamental level, reality is discrete. Math is layer upon layer of abstract toolsets to operate on integers. In that sense for me, it is very real, but invented.
Pi is no more of a human construct than any of the integers. Pi is inherent in Nature as are the integers, but more mysterious. If you accept the existence of
the integers then infinity exists and therefore pi exists too; it does not need to be constructed.
There may be no circular objects, but thats mostly because the universe isn't two-dimensional. There certainly appear to be spherical objects (to varying degrees of approximation), and pi pops out quite naturally when you ask questions about the geometry of such fundamental objects as a black-hole.
The fact that reality can be described by calculus would suggest that reality is continuous and not discrete. But I haven't seen any real evidence of either claim.
This point of view is conflating two meanings of the world "real".
The "real" in "real numbers" has ultimately not much to do with our everydays notion of real. I'd rather treat it as an arbitrary name. You could as well call them "asdfasdf numbers" and they'd remain the same.
This book is not really addressing the more common "is math real" question of it being empirical or invented. For an interesting take on that question, see the 1st section of the 2nd part of Daniel Shanks' Solved and Unsolved Problems in Number Theory. He makes some interesting points about the old Pythagorean views
It's interesting that mathematicians, when asked philosophically, might have all kinds of interesting and nuanced ideas about this topic. But when you let them get back to their mathematics, they behave as if they believed deep down in their heart that mathematics exists independently of the observer.
(It's a working attitude that works well in practice. Just like a heliocentric world view works well enough for most celestial navigation you can do without computers.)
Sorry to break it to you, but most mathematicians don't give a s*t. Even worse, a large percentage of mathematicians (not sure if "most", but I'm afraid yes) do not usually have "interesting and nuanced ideas" (nor opinions) on anything.
> This book is not really addressing the more common "is math real" question of it being empirical or invented.
Please note, this is mentioned at the beginning of the review:
"I settled in to read the book “Is Math Real?” expecting to become embroiled in the age-old controversary of whether math is invented or math is discovered. Instead, I found myself confronted with two viewpoints of mathematics: one view is that mathematics is a stiff and fixed set of rules and algorithms while the other view is that mathematics is flexible and our understanding of math comes from questioning of why mathematics functions so effectively.
The premise of “Is Math Real?” is that people have different emotions about math. Some love the math and have little difficulty determining the correct answer to a problem while others loathe and dislike the math and have a difficult time ascertaining the correct response. Many times, a student is humbled or chastised for asking ‘a stupid question’. Author Cheng states that there are no stupid questions. In fact, the most profound concepts in mathematics are learned from asking the simplest of questions.”
For me, both questions "is math real" and "is math discovered or invented" miss the point. Math is a model of the universe in the same sense that a world map is a model of the earth.
Is a map real? Well, it is. I can see it on my desk. Is the earth real? It is too, but they are not the same. In that sense map is also not "real".
Is the map discovered? Well, it uses data that was mostly discovered, but some parts were "invented" or edited for simplification for the map to be useful.
The real question should be "is math useful" as a model. We all know most basic parts are, but some mathematicians forget that they are dealing with an imperfect model and keep finding paradoxes. It's like we would forget the imperfections caused by the mercator projection and be surprised the real world distances are not proportional to map distances.
That's the reason I always liked engineering more than maths. When programming you always "import" the libraries you need and find useful for the task. You only make sure that they are compatible with each other. Mathematicians "import" all axioms, call them maths, and are surprised they get paradoxes.
I think the question should go even deeper. There are so many fundamental axioms that must be accepted on faith alone. The question I usually start with is "Can anyone prove that numbers exist outside of our imagination?" I not talking simply about perception. Even I believe that if I perceive that I am hit with a brick then the brick exists. We have no senses that can detect numbers. When I asked this question to any of the several mathematicians that I know, the answer has always been ~ Yeah, good question ~ and then they move on.
And I would dare to disagree right here. Math contains many structures that we don't know from our universe and that probably do not exist in our universe. If math is a model of universe, why is there a Mandelbrot set?
Yes, that is an intrinsic property of all models. They are imperfect, and we accept it as long as the models are useful for some purposes.
My map has a text written on it saying “Pacific Ocean”, yet I would not complain if I went to this place an couldn’t find a giant object in the ocean that would look like a letter P from the skies.
> Math is a model of the universe in the same sense that a world map is a model of the earth.
Except math can hypothetically model any consistent universe, not just our universe, which kind of undercuts the argument that it uses data that was mostly discovered, or that it's merely a model.
I think the most general view is that math is the study of structure, and some structures are real (in the sense that they exist in our universe), and some are not but we can still "discover" them by selective permutation or enumeration of axioms.
Cartography can also model any consistent universe, and I fail to see how that changes anything for the “it’s just a model” argument.
We can permutate and enumerate symbols for mountains, rivers and roads on a piece of paper. Maybe we would even get some “interesting” results like a map of the Lords of the Rings universe. How would that change anything?
I think you are both getting lost in the weeds trying to make this metaphor work, or not work.
Math is simply the logical conclusion of a set of conditions someone accepts as inherently true. If this, then that. Follow this logic far enough and you end up where we are today.
I think this conflates logic and mathematics. Some would dispute that logic underpins mathematics. Counting can be analyzed logically, but it does not in any meaningful sense seem to depend on logic.
Sure it does. How else would you prove that one number follows or precedes another? I think you are conflating the act of physically counting with the logical foundation of our number systems.
It doesn't seem correct to equate mathematics with proof. If I express a mathematical construction like the whole numbers (let Whole = Zero | Succ Whole), and I build further constructions on that foundation, am I doing mathematics? If so, then it seems mathematics does not depend on logic, as logic depends on propositions and there are no propositions to be seen.
Certainly you can analyze such constructions using logic, but that's again conflating logic with mathematics. There's overlap, but they aren't strictly the same.
I would say that a mathematical construction is based in logic, maybe not in the traditional sense, but there is definitely logic behind the construction itself. I think we are talking past one another, what I mean by logic here is more nebulous than predicate calculus. There is an innate logic behind the philosophy of mathematics and I believe you cannot divorce mathematics from logic.
You're taking a distinct philosophical stance, but you're also being unnecessarily dismissive by claiming other stances "miss the point".
Math is nothing like a map -- maps are approximations of something real and they don't have any kind of internal consistency or complexity.
But there's a good argument that math is the fundamental nature of the universe, and mathematical discoveries lead to predictions of real-world behavior. While maps don't predict a thing.
The philosophical discussion isn't around whether math is useful for tracing the arc of a ball in the air, for which it always will be merely a useful approximation. It's more around math as the language of the universe, in things like quantum physics -- there's no "approximation" here, it's more the nature of reality itself.
And here, the philosophical questions around whether our descriptions of quantum physics are "invented" or "discovered" go quite deep, and necessarily involve the nature of human knowledge itself. For many people, these don't "miss the point" at all -- they're some of the deepest, most profoundly meaningful questions that exist.
> You're taking a distinct philosophical stance, but you're also being unnecessarily dismissive by claiming other stances "miss the point".
I read my comment again and I was surprised, as I did not intend this tone. I’m sorry for being dismissive and for generalising too much about mathematicians.
Could you elaborate or point me to a formulation of the “language of the universe” argument you mentioned that avoids mentioning quantum physics? I don’t understand quantum physics and I’d like to avoid falling for the quantum physics fallacy [1]
And none of this has anything to do with the "quantum physics fallacy" at all. Philosophically, it's simply an argument about the most basic physical understanding of our universe, and right now that happens to be quantum physics.
> It's more around math as the language of the universe, in things like quantum physics -- there's no "approximation" here, it's more the nature of reality itself.
Why is it that everyone thinks of mathematical models of quantum mechanics as much closer to the "nature of reality" than any other mathematical model? If anything the constant disagreements between quantum mechanics and physical models at other scales should make it clear that all the models we have are wrong by virtue of incompatibility.
> Math is a model of the universe in the same sense that a world map is a model of the earth.
That describes pre-1900s math we inherited from the greeks. With advent of non-euclidean geometry and abstract math, math is no longer bound to objective 'reality'.
Math is invented with a purpose to help humans understand the economy and the world around them. Economy, in turn, was also invented to help humans organize their resource use.
The more humans understood the world, the more they tried to apply math and other sciences (also invented by humans) in order to explain it.
It's not even a question. Two apples will always be two apples. It's just that, without math, it would be "an apple and another apple next to it".
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[ 5.5 ms ] story [ 528 ms ] threadSee https://computingbiology.github.io/docs/hamming1998.pdf "Mathematics on a Distant Planet" for more thinking. Many physicists think that math is instrinsic and absolute (objective) in the universe, not "merely a mutual agreement"
Except circles don’t exist in the real world. They are an abstraction that mankind created that are similar enough to many natural objects as to be a useful abstraction of them.
In order for two alien races to communicate about pi, they’d have to first agree on the definition of a circle as a set of points equidistant from a central point.
That’s one of the axioms they’d have to agree on before they could even get to pi.
Then again, maybe a circle (and all other geometric abstractions) are too tied specifically to visual perception. Maybe, for example, intelligent aliens who only have auditory perception, would have no concept of a circle but would have gotten to the Doppler effect. Or intelligent aliens who only have tactile perception, would have no concept of a circle but would have gotten to the laws of thermodynamics.
(1) Ratios between things in the world.
(2) Logical relations between things in the world.
(3) Absolute distinctions between things in the world, in a nominative sense, which is to say in the sense that numbers can be assigned to things or elements of things.
Mathematics that relates to one of the three use-cases above is absolutely real, hence its unreasonable effectiveness in the natural sciences. (See Wigner: https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf )
When mathematics does not relate to one of the three -- for e.g., in Cantor's theories of "countable" and "uncountable" infinite sets -- it is totally unreal. A construct or game played with logic that has no prior or intrinsic relevance to the material world.
Is realness not an instrinsic property but just a judgement on how useful something is?
Is being real a real property of something or just a construct?
It misses the point of the question "is math real"?
Tbf, i am actually sympathetic to the position that "realness" is not really relavent or a well defined term when it comes to what is basically a descriptive language of patterns and relations. But "is it real" is the question we started the thread on.
Whence the pragmatic maxim:
Pragmatically speaking, math is no more or less real than that.The universe, as we know it, is simulatable. In other words, it can apparently be reduced to mathematics in precisely those three senses I outlined above. (If it helps, you can imagine electrons and other subatomic particles as bits of information in a coordinate space, which are constrained to operate in accordance with rules that govern logical relations between things.)
What mathematics is real? Anything that would relate to that universe or any of its constituent parts, in _any_ meaningful sense. We don't need to have found a use for it, and surely there's a great deal that is still undiscovered.
What mathematics is unreal? Whatever is, a priori, absolutely unnecessary to the existence of such a universe -- or, even worse, would break a universe described in terms of logic were it somehow made manifest.
Are you considering the set of natural numbers to be “unreal”? Or, if you consider the set of natural numbers to be real, do you not consider power sets to be a logical relation?
After all both of them are mostly full of things we will never need, encounter, or even be able to define.
As for this:
> Or, if you consider the set of natural numbers to be real, do you not consider power sets to be a logical relation?
Skolem solved that quite neatly already: Every "uncountable" set has a countable model. Thus the power set is, in fact, no larger than the set of natural numbers, because both can be fully described in a countable manner. What I'm describing is necessarily an abstraction of an abstraction, though, so I'd consider it "unreal" by definition.
I’m not quite sure what a “model of a set” means. I suppose maybe you mean like, the set in some model, which is described by the given description of a set in the theory?
> Thus the power set is, in fact, no larger than the set of natural numbers, because both can be fully described in a countable manner.
I don’t think this follows.
There are countable models of set theory. And for these models, there is (in the meta-theory) a bijection between the set representing the set of real numbers, and the set (in the meta theory) of natural numbers.
I don’t think this establishes that “in fact” there is a bijection between the real numbers and the natural numbers.
Rather, in any model of any of the usual set theories, there will be no bijection between the set of real numbers and the set of natural numbers. (Of course there will not be, because these theories entail that there is no such bijection.)
If I take a non-standard model of arithmetic, and for some non-standard natural number n, and consider the uniform distribution of (non-standard) natural numbers less than n, then for every standard natural number, the probability of getting that natural number from that distribution, will be equal. Are we therefore to conclude that there is a uniform probability distribution over all the standard natural numbers? By no means!
Preliminaries: (1) Let T be a first-order theory with a countable or uncountable infinite model M. (2) Let L be the language of T. (3) T is assumed to be a set of sentences (closed formulas) in L. Algorithm Steps:
Initialize Countable Set S: Start with S = Const(L), the set of all constant symbols in L.
Extend Language: For each n-ary relation R in L and each n-tuple (c1, ..., cn) of elements in S, if M⊨R(c1, ..., cn), then add a new constant symbol c to S and extend L to L' by adding c.
Iterate for All Formulas: For each formula ϕ(x1, ..., xn) in L and each n-tuple (c1, ..., cn) of elements in S: If M⊨∃xϕ(c1, ..., cn), then pick some a in M such that M⊨ϕ(a, c1, ..., cn). Add a new constant symbol c to S to represent a and extend L to L' by adding c.
Closure: Repeat Steps 2 and 3 until S no longer changes. Since T and L are countable, this process will eventually result in a countable set S. Construct Countable Model N: Take the substructure of M generated by S as N. By construction, N is countable.
Elementary Submodel Check: By construction, N is an elementary submodel of M. This is because for any formula ϕ and any n-tuple (c1, ..., cn) from S, M⊨ϕ(c1, ..., cn) if and only if N⊨ϕ(c1, ..., cn).
Conclusion: N is a countable elementary submodel of M, and therefore T has a countable model. End.
You're right about the bijection, but, in the countable model, the set that "represents" the real numbers is countable. (From the perspective of the meta-theory.) However, within the model itself, this set still satisfies all the axioms that make it "look" uncountable. For example, there's no bijection between this set and the set representing the natural numbers within the model, even though such a bijection exists in the meta-theory.
This is why Skolem's solution can be most succinctly described: "Every set has a countable model if one steps outside the set and constructs a countable model from its elements."
In Kleene's “Introduction to Metamathematics,” he describes the situation as follows: "Either we must maintain that the concepts of an arbitrary subset of a given set, and of a non-enumerable set, are a priori concepts which elude characterization by any finite or enumerably infinite system of elementary axioms; or else (if we stick to what can be explicitly characterized by elementary axioms, as we may well wish to in consequence of the set-theoretic paradoxes ) we must accept the set-theoretic concepts, in particular that of non-enumerability, as being relative, so that a set which is non-enumerable in a given axiomatization n may become enumerable in another, and no absolute non-enumerability exists."
I find that to be quite a bold statement. The question as to whether "mathematics is real" feels firmly rooted in philosophy. First, I would question what is meant by real and unreal. Second, I would question if the answer matters whether the existence of the real and the unreal are not yet discovered, or even imagined.
Alright great lol
Think of a triangle. Now draw that triangle on paper. If you look closely enough, you'll see "imperfections" in the triangle you just drew. Now ask yourself: "how do I know this thing I just drew is imperfect? Where did the idea of a perfect triangle come from?"
Plato would say the perfect triangle comes from the realm of "forms". This mystical place which is "more real" than "reality" because everything there is perfect and everything here is just a flawed approximation. Plato also said that this is the place where our souls go when we die, and we engage in "congress" with the forms and then return to earth, reincarnated. When we learn things, we aren't learning something new but actually recalling memories of the forms. This is why everyone knows what a perfect triangle is but no one has ever seen one in the physical world.
> CHENG: I'm not trying to answer whether math is real or not. I'm trying to show that considering the question at all leads us to interesting thoughts. And in the end, what I say is that, with all these questions, I don't think there are yes-or-no answers, and we shouldn't claim that there are. What we should do instead is say there is a sense in which - you know, what is the sense in which math is real, and what is the sense in which math isn't real?
> And the thing is, I think a lot of people who say math isn't real are using that to say, oh, so it's irrelevant and stupid. Why should we study it? It's made up. And what I want to say is that just because it is made up doesn't mean that it's irrelevant. And actually, the fact that it's kind of made up makes it really powerful because - well, it makes it really, in a way, more accessible because you don't need a lot of money to get it. All you need is an imagination. And I think that's a really amazing thing about it. And just like fiction isn't real, but fiction can give us insights about the world around us to highlight much more, specifically, things about society. And that's what I think is powerful about abstract math as well - because we're not constrained by reality.
I haven't read the book, but it seems like the title may be a throwaway question meant to pique the reader's interest and say "let's get philosophical about math".
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[1] https://www.npr.org/transcripts/1193035114
A quadratic equation can be used to approximate the coordinates of the trajectory of a thrown object in a gravity field. I think the really interesting question is why that particular equation reflects that trajectory. Arithmetic operators - multiplication and addition in that case - in a particular order, are approximating the causal operations of existence that are actually at work. I think of this as the philosophy of mathematics and much more should be done to investigate it.
The mathematics to accurately predict or relay reality is still complex enough that it’s often beyond us. You’re right in that it’s a tool to understand, but if we’re using simplified math for simplified reality, is it really epistemological?
As you suggest, math isn’t outside the boundary of philosophic investigation. It never was in the past, and I don’t think better approximations change that calculation.
How can any perception be independent of mind consciousness? And how can mathematics reflect anything other than that consciousness?
"Within consciousness" doesn't belong here. Even if all sentient life in the universe died out tomorrow, the idea of triangles (and "triangluarity") would still exist.
this "the idea of triangles would still exist" statement looks very reasonable, but can we ever proove it? you can not remove ALL consciousness to verify. at least a bit consciousness must be in the system to do the experiment.
yes, it very much coincides with all of our understanding of reality, that abstract ideas like triangluarity is independent from any consciousness. at least any consciousness known or conceivable to us so far.
so my extension to this statement would be - admittedly supernaturally-sounding - that: Even if all sentient life in the universe died out tomorrow, the idea of triangles (and "triangluarity") would still exist, possibly because there is a consciousness above/beyond of the conceivable the universe to sustain their existence.
If so, then consciousness doesn't need to exist for information structures to exist.
No, I'm not a Billy Joel expert or fanatic. I just happened to notice, all right?
We're not in school.
I was surprised later in life when the majority of people I talked to felt Math was this way (and a good thing). To them, math is a set of rules you learn to follow to the T and use them on other problems.
For me, everything a Math teacher conveyed was more of a recommendation, a suggested tool that I could incorporate into my tool box. The methods they employed to solve problems were a matter of preference to me, rather than rigid rules.
This way of thinking has always had some pros and cons. I never solved a problem the way a 'grader' was expecting. Some teachers loved the creativity and efficiency, other TA's just marked as zero. It made applying what I learned to other things, but on exams I would always be stressed with time because I would spend time on which way I was going to answer the problem.
That being said, I am very good at Math. scored well in HS/college, 165/170 Math GRE score, in a field where Math is important (Data Scientist).
Define real.
Is math real or not? It doesn't matter she posits, it works and continues to work and we can learn from its existence that almost everything can be explained and is "predictable" given enough inputs.
> 3) math has an indirect usefulness which is a way of thinking that is transferable to a myriad of disciplines and solutions to problems in everyday life. And it is this third reason that makes math relevant for most people.
Are there actually any good studies that show this in a counterfactual setting? Like, do we actually know that spending time teaching maths (or, less plausibly, Latin) helps students aquire these abstract skills more than other subjects? Is this "transferability" of meta-skills a testable outcome?
My feeling is that the utility of higher math, Latin, history and literature comes every minute of every day, as you experienced life as someone who has familiarity those things and your life will be richer and fuller.
This is decidedly not testable. And yet I still believe it.
The skill of logical inference — even at the level of very basic syllogisms — is both very much underappreciated and underdeveloped in the American college population, at least from my personal experience. As good citizens, we all collectively should grab a couple of Martin Gardner's or Lewis Carroll's books off the shelf and give them a good read. I predict it will do much good... and if I'm wrong, it certainly won't do any harm!
I took logic in college, and although the "logic as english statements" stuff was sort of confusing, the symbolic stuff like A&B = !A|!B stuff has helped with computers all my life.
It was only much later in life that I ran back into logic as english statements in a way that made sense as practical.
I read a book where they took apart the statement:
Most people in relationships will respond to this with: But the book explained that with "If X, then Y" it was futile to address Y. You must address X: ...just hard to unpack this in the middle of an emotional situation unless you've studied the logic:)
https://www.youtube.com/@thinkingmathematically
It might help to answer the question
Going through proofs and proving things on your own really transferred to being able to better present arguments. The diversity of the math I learned has helped to reflect on things from different perspectives.
In sum this helped with everything from thinking more and better about the core issue at hand, writing argument chains in the correct order, cutting down on irrelevant stuff and more.
I've used this to significantly help the grades of both my SO and a family member, who both took non-math topics, by improving their hand-ins. I didn't know their field so was strictly improving the structure and presentation, and asking for clarifications where I felt the arguments didn't add up, and have them write down the answer.
I feel it still helps me a lot writing emails at work and similar.
Basically, I believe that there's a heavy correlation between being good at maths and being good at solving every day (and not so every day) problems.
But I don't believe there's much of a correlation between being taught math at school for even more hours will make much of a difference. Most schools are terribly at teaching anything.
And, since LLMs are so bad at math currently, we may find that, by improving their math ability with gobs of synthetic data, we get improvements in general reasoning.
One aspect that's quite easy to criticise about this study is that it uses existing groups of students with different levels of maths training. This means that there is possibly self-selection etc, and one may argue there might also be a causal effect in the opposite direction (e.g. folks that are good at reasoning like to do maths.)
We know that if you try to train someone in math (or in Latin), and they do well, then they will also do well at other things in the rest of their life. Some people would like to give the credit for that good performance to the Latin training.
Is there anything that physically exists that is non-mathematical?
There are real limits in what humans know about the universe. But it's possible in theory that everything that is "real" follows from some differential equations.
Could your question be rephrased as "take the real world, with its things we don't have an accurate model of (which is most of them). Would we notice if we replaced this universe by a tidier one, perfectly described by a math model?".
It's an interesting, scifi question alright. I don't know if it has an answer.
First, what would our understanding of mathematics improving look like? Mathematics is an abstraction. Does such a thing as "true" mathematics even exist?
Second, what evidence is there that improving our understanding of maths will asymptotically improve our models of reality? There really is no evidence of this! Seems like a wild logical leap.
The map is not the territory.
But are you sure about this? If mathematics were not prescriptive then we should expect to see some evidence of that. So far nobody has demonstrated the existence of phenomena that contradicts the equations of physics. Usually we would call such phenomena "supernatural."
It's possible in theory that nature doesn't have to obey mathematical laws. But if there's no evidence for it and if it doesn't explain any phenomena, then what is the argument for holding that particular belief?
Math is a collection of abstractions. Some are useful in explaining natural phenomena. Some are not.
You cannot contradict a description; it's not a "law" in the legal/prescriptive sense.
How would you even "contradict the equations of physics"? The real world exists and it's there; we use models to imperfectly describe (to varying degrees of success) its phenomena. In a sense, the real world contradicts those equations because they do not perfectly describe it; they are just an approximation.
That's kind of a tautology because physics is rules for what we observe. If we observe something contradicting the rules then we come up with new rules to describe the new thing.
That's basically asking if there's true randomness.
Philip Davis, Reuben Hersh, "The Mathematical Experience"
Not when we are talking about how Math can be Complex.
[1]: Galileo says that, roughly, the book of nature is written in the language of mathematics https://en.m.wikipedia.org/wiki/The_Assayer
[2]: A classical essay by Wigner on the "unreasonable effectiveness of mathematics" also brings up the language analogy. https://web.archive.org/web/20210212111540/http://www.dartmo...
Are both language and math a third thing? Like maybe automata?
> The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed.
Translates to
> F=ma
Mass is represented in English by the tokens: "the property of matter resistant to acceleration"
Acceleration is represented in English as the change of the change of the position through time.
To your point, there are an infinite number of definitions we can apply to the equation defining a product. So Physics context + mathematical verbiage = elucidated ideas.
I'm also not sure that translating without context is a meaningful. What language operates without context? Without context, what would the pronoun 'that' in your post point to? What does the phrase 'back to' mean without the associated physical metaphor? How would this be different from translating to and from German without context?
Force = velocity * resistance
Sorry, if this isn't clear to everyone, why does this result in the rejection of mathematics as a language rather than the rejection of the question of the realism of mathematics? The latter seems much easier with no downsides (that I can see), whereas the rejection of mathematics as a language has many downsides.
Are languages real? Sure, I guess. Why not. They’re made of concrete sounds and symbols that everyone (by design) can point to - they might even be the realest thing there is.
Are (math) concepts real?
As in like, if I rearranged all the matter that exists out there, would the value of root(2) stop being irrational?
Or maybe they don’t have to be physical to be real; maybe being “real” has more to do whether or not something has an effect on something else than its physical existence. Then, there’s a difference between how something exists and how real it is?
Etc.
Thinking of math as a language as opposed to a set of concepts isn’t interesting to the discussion of its reality IMO
2. While I would say that most would trivially agree that mathematics is a form of language [2], it is less common to argue that mathematics is _just that_. And, crucially, it is not supported by neuroscience [3]. It seems that there are distinct brain areas involved specifically in mathematical thinking outside of those that are language-specific, which would mean that mathematical thinking is more than language [3].
[1] For a criticism regarding galileo's mediocrity and historically unfounded idealisation https://intellectualmathematics.com/blog/the-case-against-ga...
[2] Actually, a collection of languages, as different parts of mathematics form different languages or dialects that are not easy to communicate in between even if they refer to similar objects, eg category theory vs set theory, analysis vs probability theory.
[3] https://anthonybonato.com/2017/09/19/this-is-your-brain-on-m...
I would not construe a small study of 30 people (15 math experts) as having the backing of the field of neuroscience.
The steel man position of mathematics-as-language would be something along the lines of "Mathematics is mostly a language + some other stuff". The paper in question only shows there is "some other stuff"---but that is also true of almost every non-contrived use of language. The paper shows that the other stuff for sentences about history and nature appear differently in the brain than the other stuff for sentences about math, but that is far from demonstrating that math on the whole is not mostly a linguistic phenomenon.
Personally, I think case studies of famous mathematicians or physicists discussing their thinking process (e.g. Einstein) is more convincing evidence against the hypothesis than fMRI studies.
But to me it seems that even when you consider math as just being language… you still run into the same philosophical problem when you consider what the semantics of that language are.
For example, you may believe that the practice of algebra is simply operations on stings of a language. But when you start asking what objects those sentences refer to, you end up asking what math is again.
Math has no fundamental non-differentiable epistemology qua mathematics.
As you said there is no meaningfully discrete, bounded “object” that can be used as a universal reference - so the fact of relativity means that even if maths were “real”
I’m not even sure such a thing exists (who says the rules of the universe are uniform, or static as we’re only have a few centuries of poor observation) and if it does exist, if humans have the ability to identify it
"What can math describe" is an easier question than "what is reality?" or "what is the nature of all concepts a human mind can conceive of or be interested in?". Not that it's an easy question either.
I hope all these books, YouTube videos, and websites can help in making students curious about mathematics and explore further on their own. It is hopeless to even expect the school systems in US and a lot of other countries, to change the way their school systems work - teachers are not given enough time to spend on the topics in mathematics. It takes lot of extra effort by the students.
Edit: I thought it was pretty advertent.
The bits you need from calculus to approach those other subjects is also very minimal and approachable relative to the content of a calculus pre-req (where applicable).
Because ℚ ⊂ ℝ of course, but it is also not that far off from the conclusion of the book in that "math is real because it is an idea and ideas are real". In a non-mathematical sense, rational means "based on reason", and reason is the power of the mind.
But can math be both rational AND irrational?
There is nothing that says that the distribution between rational and irrational numbers that show up in nature is the same as the distribution in our construction of the real numbers.
It's a bit more complicated than that.
Similar, an electron in a single isolated atom has specific quantised energy levels. But if you look at the electrons in a hunk of copper, they are essentially free to absorb and emit energy in almost arbitrary amounts.
An even stronger example is time: as far as I am aware, time is not quantised in any of our accepted theories.
There's some reasonable speculation that ultimately everything is quantised at the Planck scale, including time. But that's just a very reasonable hunch, not something that 'physics has shown'. (And you already point out that trying to marry quantum mechanics with general relativity is a hot mess.)
I agree that 'quantisation all the way down' is the way to bet. But that's just speculation.
(But I strongly disagree with your claim that physics is build on integers. Yes, it might be discretised, but there are plenty of discrete structures that are not integers. Look at a Rubik's cube for a simple example. On top of that: almost any real world measurement is better described by a probability distribution than by single number, be that an integer or otherwise.)
This is not the case for irrationals... therefore it is concluded that the infinity of irrationals is a larger infinity than the infinity of rationals.
See:
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
https://mathworld.wolfram.com/CantorDiagonalMethod.html
That is true, but it's never taught. I don't even know what that mapping is, though I've seen it mentioned once in a popular treatment.
What's taught is always the mapping from naturals to rationals that overcounts the rationals, hitting them all an infinite number of times. (Because it's very easy to show a bijection between the naturals and the ordered pairs, but while (2,3) and (4,6) are distinct ordered pairs, they do not represent distinct rationals.)
But then all you've shown is that the naturals are at least as numerous as the rationals. To show that the naturals and the rationals have the same cardinality, you either rely on the idea that the naturals are the smallest infinite set, or you appeal to the fact that the naturals are a subset of the rationals.
My favorite way of visualizing the difference uses the fact that every rational has a repeating decimal after some nth decimal place, and no irrational has a repeating decimal. Say you want to construct a number x, where 0 < x < 1, by drawing integers 0 through 9 randomly from a hat. Each integer drawn from the hat is placed at the end of the decimal; for example, if you draw 1,3,7,4 then the decimal becomes 0.1374. You then draw, say, 1, and it becomes 0.13741, and so on. If you could draw infinitely many times from the hat, what is the probability that you'll construct a number with a repeating sequence? That would give a rational number.
Indeed, you've grasped the core of it. There's no rule you can write for irrational numbers such that "b is the next number after a", because there are infinitely many numbers between a and b that you'd be missing. You can't count them, i.e. you can't map them to integers.
Uncountable Infinities > Countable Infinities
The argument as to why the irrational numbers are uncountable and the rationals are countable is more involved than what you've made out. But very simply you can think of it as you need an infinite string of digits to describe each irrational number, but each rational number can be written as two finite strings of digits (in the form A/B, where A and B are integers). So to write our the irrationals you have an infinite number of strings, where each string is also infinitely long, while with the rationals you have an infinite number of strings, but each string is finite.
That's literally the same thing. What is counting if it isn't being able to say what the next thing is? Do you have a mapping to integers or not? If so, then every n has n+1.
I know it was more complicated, but jaza had the essence of it. Without what they observed the whole thing falls apart. Yeah, it still needs proof, but I'm pretty sure five other comments went there.
> So to write our the irrationals you have an infinite number of strings, where each string is also infinitely long, while with the rationals you have an infinite number of strings, but each string is finite.
You've set the table but forgotten the feast! You're missing the step where you demonstrate that there's a number that isn't in this list. (Hint: think diagonally.)
The point I was trying to make is that there is no concept of 'next' inherent to the rationals, nor is there any natural or canonical ordering. The ordering and what comes 'next' is entirely a property of which arbitrary mapping you choose (I'm partial to Gödel numbering). The resultant order that your mapping imposes on the rationals is rarely useful or meaningful.
That ordering doesn't have the property that all sets of rationals contain a least element, or that any rational has a successor rational. (That would make them "well ordered".) But it's a natural ordering.
>> The argument as to why the irrational numbers are uncountable and the rationals are countable is more involved than what you've made out. But very simply you can think of it as you need an infinite string of digits to describe each irrational number, but each rational number can be written as two finite strings of digits (in the form A/B, where A and B are integers). So to write our the irrationals you have an infinite number of strings, where each string is also infinitely long, while with the rationals you have an infinite number of strings, but each string is finite.
This argument doesn't actually work. If there were only a countable number of irrational numbers, you could specify them all fully by doing no more than a countable amount of work, even stipulating that describing a single irrational number requires listing a countably infinite number of digits.
> Indeed, you've grasped the core of it.
What? That's not the core of anything. It tells you that the irrationals are dense in the real number line. You know what other set is dense in the real line? The rationals.
Mathematicians can even meaningfully compare infinities.
See eg https://www.cantorsparadise.com/this-may-seem-more-irrationa... or https://math.stackexchange.com/questions/474415/intuitive-ex...
You can also look at eg a uniform random variable on the interval between 0 to 1. The probability of hitting a rational number is 0%. The probability of hitting an irrational number is 100%.
> Or is the reasoning that, because there is an infinite quantity of irrational numbers between any two given rational numbers, there are therefore many more irrational numbers than rational numbers?
No, that's not enough. There are also an infinitely many rational numbers between any two given irrational numbers.
Well... one of the consequences of that precision is the theorem that there is no such concept as choosing a real number "at random".
(Sqrt(2) as a real number, is actually encoded as the set of all rationals less than sqrt(2) on the number line).
There are incomputable models.
Pi shows up in many physics equations, but that’s entirely due to our choice of units.
The "real" in "real numbers" has ultimately not much to do with our everydays notion of real. I'd rather treat it as an arbitrary name. You could as well call them "asdfasdf numbers" and they'd remain the same.
(It's a working attitude that works well in practice. Just like a heliocentric world view works well enough for most celestial navigation you can do without computers.)
Please note, this is mentioned at the beginning of the review:
"I settled in to read the book “Is Math Real?” expecting to become embroiled in the age-old controversary of whether math is invented or math is discovered. Instead, I found myself confronted with two viewpoints of mathematics: one view is that mathematics is a stiff and fixed set of rules and algorithms while the other view is that mathematics is flexible and our understanding of math comes from questioning of why mathematics functions so effectively.
The premise of “Is Math Real?” is that people have different emotions about math. Some love the math and have little difficulty determining the correct answer to a problem while others loathe and dislike the math and have a difficult time ascertaining the correct response. Many times, a student is humbled or chastised for asking ‘a stupid question’. Author Cheng states that there are no stupid questions. In fact, the most profound concepts in mathematics are learned from asking the simplest of questions.”
Is a map real? Well, it is. I can see it on my desk. Is the earth real? It is too, but they are not the same. In that sense map is also not "real".
Is the map discovered? Well, it uses data that was mostly discovered, but some parts were "invented" or edited for simplification for the map to be useful.
The real question should be "is math useful" as a model. We all know most basic parts are, but some mathematicians forget that they are dealing with an imperfect model and keep finding paradoxes. It's like we would forget the imperfections caused by the mercator projection and be surprised the real world distances are not proportional to map distances.
That's the reason I always liked engineering more than maths. When programming you always "import" the libraries you need and find useful for the task. You only make sure that they are compatible with each other. Mathematicians "import" all axioms, call them maths, and are surprised they get paradoxes.
What do you mean by "exist" here?
And I would dare to disagree right here. Math contains many structures that we don't know from our universe and that probably do not exist in our universe. If math is a model of universe, why is there a Mandelbrot set?
My map has a text written on it saying “Pacific Ocean”, yet I would not complain if I went to this place an couldn’t find a giant object in the ocean that would look like a letter P from the skies.
Except math can hypothetically model any consistent universe, not just our universe, which kind of undercuts the argument that it uses data that was mostly discovered, or that it's merely a model.
I think the most general view is that math is the study of structure, and some structures are real (in the sense that they exist in our universe), and some are not but we can still "discover" them by selective permutation or enumeration of axioms.
We can permutate and enumerate symbols for mountains, rivers and roads on a piece of paper. Maybe we would even get some “interesting” results like a map of the Lords of the Rings universe. How would that change anything?
Math is simply the logical conclusion of a set of conditions someone accepts as inherently true. If this, then that. Follow this logic far enough and you end up where we are today.
Certainly you can analyze such constructions using logic, but that's again conflating logic with mathematics. There's overlap, but they aren't strictly the same.
Math is nothing like a map -- maps are approximations of something real and they don't have any kind of internal consistency or complexity.
But there's a good argument that math is the fundamental nature of the universe, and mathematical discoveries lead to predictions of real-world behavior. While maps don't predict a thing.
The philosophical discussion isn't around whether math is useful for tracing the arc of a ball in the air, for which it always will be merely a useful approximation. It's more around math as the language of the universe, in things like quantum physics -- there's no "approximation" here, it's more the nature of reality itself.
And here, the philosophical questions around whether our descriptions of quantum physics are "invented" or "discovered" go quite deep, and necessarily involve the nature of human knowledge itself. For many people, these don't "miss the point" at all -- they're some of the deepest, most profoundly meaningful questions that exist.
I read my comment again and I was surprised, as I did not intend this tone. I’m sorry for being dismissive and for generalising too much about mathematicians.
Could you elaborate or point me to a formulation of the “language of the universe” argument you mentioned that avoids mentioning quantum physics? I don’t understand quantum physics and I’d like to avoid falling for the quantum physics fallacy [1]
[1]: https://www.logicallyfallacious.com/logicalfallacies/Quantum...
And a good place in general to start is always Wikipedia:
https://en.wikipedia.org/wiki/Philosophy_of_mathematics
And none of this has anything to do with the "quantum physics fallacy" at all. Philosophically, it's simply an argument about the most basic physical understanding of our universe, and right now that happens to be quantum physics.
Why is it that everyone thinks of mathematical models of quantum mechanics as much closer to the "nature of reality" than any other mathematical model? If anything the constant disagreements between quantum mechanics and physical models at other scales should make it clear that all the models we have are wrong by virtue of incompatibility.
That describes pre-1900s math we inherited from the greeks. With advent of non-euclidean geometry and abstract math, math is no longer bound to objective 'reality'.
The more humans understood the world, the more they tried to apply math and other sciences (also invented by humans) in order to explain it.
It's not even a question. Two apples will always be two apples. It's just that, without math, it would be "an apple and another apple next to it".