Isn't this article conflating our formalism of a given abstract entity (like real numbers or integers) with the abstract entity itself? Surely quantities existed long before humans (e.g. there was a quantity of stars in the Milky Way 1 million years ago). And surely ordinals existed long before humans (e.g. there was a most massive star in the Milky Way 1 million years ago).
The article's claim seems to be about the mathematical formalisms humans have invented for integers and real numbers. And I agree that our formalism of integers is simpler and more elegant than our formalism of real numbers. But that could just be because we've done a worse job formalizing real numbers!
There is no fundamental unit of "star". Maybe we can talk about electrons or protons or something, but what is and is not a star is a model, not a reality.
Concretely, a bundle of pre-stellar gasses at some point transitions to being a star, but when in that time spectrum does it make that transition? When in the process of stellar exhaustion does it stop being a star?
Since quantum uncertainty (in a finite universe) basically says that you can't measure anything with infinite precision - I'd argue that God created, at most, the Rational Numbers. The Reals might be the closure ( https://en.wikipedia.org/wiki/Closure_(topology) ) of the Rationals - but doing that was the work of man.
Quantum mechanics actually contains measurable real numbers (well.. complex numbers). Amplitudes are postulated to be infinitely precise, and rounding them has a tendency to introduce pretty serious consequences like FTL communication.
For example, in fault tolerant quantum computing, rotations are synthesized using sequences of 45 degree rotations around the X, Y, and Z axes. The matrices that describe those 45 degree rotations contain rational and irrational numbers (in particular: sqrt(2)). If those irrational numbers are actually truncated, this would have observable consequences. You'd need a sufficiently large quantum computer running for sufficiently long to do sufficiently accurate tomography of sequences of those rotations in order to resolve the truncation, and to be frank some of those "sufficientlies" would be very impractical to achieve especially if the truncation was small (and woe unto you if adding more qubits somehow reduces the amount of truncation!), but in principle it'd be possible.
I infer from footnote 10 that an unspoken subtext of this is that footnote 1 is that while the reader may choose a (simplistic) atheist's formulation of the idea, the author does not,
which would be consistent with their interest in the question of the "divine" and human reasoning at all, especially as argued about by theologically inclined philosophers much admired by Judeochristians.
That subtext being, discovering that our models or knowledge are incomplete somehow increases the territorty of what he's calling mysterious. By which I take it he means, knowable to and to not beat around the burning bush, attributable to the divine. By which I take it for him that he means a Judeochristian god.
One of the great and persistent bemusements of my adulthood is discovering that other adults take their religiosity not just seriously but central to their understanding of themselves, and their context generally.
It's a relief that such people have participated in construction of a society within which such beliefs are considered personal, as it saves a lot of embarassment for people such as myself, who find such notions wince-inducing, and, both their origins and utility quite transparent.
The reals aren't algebraically under multiplication; a simple equation like x*x=1 can't be handled in real numbers. The complex numbers are algebraically closed. So I suspect that God created the complex numbers.
God certainly had a fondness for the real subset. Measurements are real scalars -- so much so that it really does look like God created the reals. That's what's important to us. But the fundamental laws seem to require the complex numbers (or their equivalent, like matrices), and closure under arithmetic operations really does feel like it should be a requirement for the reality of the universe.
The double entendre of "having studied Agrippa" in the footnotes is probably going to go unnoticed unless someone mentions it here. Contemporary to the cited Camillo Agrippa, fencing master, was Henrichus Cornelius Agrippa, whose collections on philosophy and occultism are much more relevant to the topic. H.C. Agrippa's work is still considered authoritative in its fields: the numbers represent the ideas, which were both created in the first moments by God. the difference between set of reals and set of integers might have a correlation to the difference between the set of all expressible concepts, and (the smaller) set of actually meaningful concepts. Maybe some computability theory could be tossed in there too.
Not all reals are expressible: to be expressible is to have a finite representation in some language. Definable numbers [1] are that (countably infinite) subset of the reals which can be expressed individually. Almost all reals are not definable and therefore cannot be individually named.
If it turns our universe is discrete (at the Plank scale), that supports the possibility that all reality may be discrete.
Which would mean integers are baked in, rationals too, but non-constructible reals (essentially all reals, given any degree of approximation) are a useful abstraction but don't actually exist in any way.
Reals are not real.
(Roughly) Equivalently: There may be no perfect circles in nature.
What is this nonsense? I don't understand. All of these are concepts were created, invented and are commonly used by Homo Sapiens. God, math, numbers, integers, rational numbers, irrational numbers, what have you. All abstract concepts brought about solely by humans.
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[ 3.4 ms ] story [ 38.6 ms ] threadThe article's claim seems to be about the mathematical formalisms humans have invented for integers and real numbers. And I agree that our formalism of integers is simpler and more elegant than our formalism of real numbers. But that could just be because we've done a worse job formalizing real numbers!
There is no fundamental unit of "star". Maybe we can talk about electrons or protons or something, but what is and is not a star is a model, not a reality.
Concretely, a bundle of pre-stellar gasses at some point transitions to being a star, but when in that time spectrum does it make that transition? When in the process of stellar exhaustion does it stop being a star?
For example, in fault tolerant quantum computing, rotations are synthesized using sequences of 45 degree rotations around the X, Y, and Z axes. The matrices that describe those 45 degree rotations contain rational and irrational numbers (in particular: sqrt(2)). If those irrational numbers are actually truncated, this would have observable consequences. You'd need a sufficiently large quantum computer running for sufficiently long to do sufficiently accurate tomography of sequences of those rotations in order to resolve the truncation, and to be frank some of those "sufficientlies" would be very impractical to achieve especially if the truncation was small (and woe unto you if adding more qubits somehow reduces the amount of truncation!), but in principle it'd be possible.
which would be consistent with their interest in the question of the "divine" and human reasoning at all, especially as argued about by theologically inclined philosophers much admired by Judeochristians.
That subtext being, discovering that our models or knowledge are incomplete somehow increases the territorty of what he's calling mysterious. By which I take it he means, knowable to and to not beat around the burning bush, attributable to the divine. By which I take it for him that he means a Judeochristian god.
One of the great and persistent bemusements of my adulthood is discovering that other adults take their religiosity not just seriously but central to their understanding of themselves, and their context generally.
It's a relief that such people have participated in construction of a society within which such beliefs are considered personal, as it saves a lot of embarassment for people such as myself, who find such notions wince-inducing, and, both their origins and utility quite transparent.
God certainly had a fondness for the real subset. Measurements are real scalars -- so much so that it really does look like God created the reals. That's what's important to us. But the fundamental laws seem to require the complex numbers (or their equivalent, like matrices), and closure under arithmetic operations really does feel like it should be a requirement for the reality of the universe.
[1] https://en.wikipedia.org/wiki/Definable_real_number
Which would mean integers are baked in, rationals too, but non-constructible reals (essentially all reals, given any degree of approximation) are a useful abstraction but don't actually exist in any way.
Reals are not real.
(Roughly) Equivalently: There may be no perfect circles in nature.