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God created the real numbers

31 points by EthanHeilman 1y ago

tshaddox 1y ago
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!
- john-h-k 1y ago
  > 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! Everything you can express in integers you can express in reals, but there are many things expressable in reals not possible in integers. It would be surprising if the formalism for a thing that completely supersets another thing had an equally simple formalism
  - orlp 1y ago
    Actually in math it's very common for the more general system to be simpler. Compare for example the prime numbers with the integers, or general groups with finite simple groups and the monster group.
  - tshaddox 1y ago
    I'm not suggesting the two formalisms should be equally simple. But surely it's not controversial to claim that formalizing the reals involves much more advanced mathematics (and runs into much deeper problems) than formalizing the integers. I'd argue that this disparity is slightly surprising, given that both the integers and the reals are ubiquitous in essentially all branches and levels of mathematics.
  - xscott 1y ago
    > [...] but there are many things expressible in reals not possible in integers Are you sure there is anything we can express in the Reals that isn't an integer in disguise? The first answer might be the sqrt(2) or pi, but we can write a finite program to spit out digits of those forever (assuming a Turing machine with integer positions on a countably infinite length tape). The binary encoding of the program represents the number, and it only needs to be finite, not even an integer at infinity. Then you might say Chaitin's constant, but that's just a name for one value we don't know and can't figure out. You can approximate it to some number of digits, but that doesn't seem good enough to express it. You can prove a program can't emit all the digits indefinitely. And even if you could, is giving one Real number a name enough? Names are countable, and again arguably finite. It seems to me we can prove there are more Reals than there are Integer or Computable numbers, but we can't "express" more than a finite number of those which aren't computable. Integers in disguise.
    - tshaddox 1y ago
      It seems like you're suggesting that mathematicians replace the reals with the computables. This is a reasonable thing to try, and is likely of particular interest to constructivists. There's even this whole field: https://en.wikipedia.org/wiki/Computable_analysis https://en.wikipedia.org/wiki/Computable_analysis
      - xscott 1y ago
        > It seems like you're suggesting that mathematicians replace the reals with the computables. No, not any more. :-) I liked the idea a year or two ago, but I've come to believe that even the Integers are too bizarre to worry about. For now, I'm content just playing with fixed and floating point, maybe with arbitrary precision. Stuff I can reason about. I just think people are a little too casual thinking they are really using the Reals. It might be like Feynman's quote about saying you understand QM.
- crazygringo 1y ago
  I mean, this is one of the deepest questions of philosophy. Do or can concepts and categories exist without the beings that create them? Does tuna casserole exist independently of humans? If not, how is the idea of the number 7 different from the idea of a tuna casserole? Or what about the concept of decision by majority, which isn't as basic as 7, but doesn't have the physicality of a casserole?
  - 1718627440 1y ago
    > Do or can concepts and categories exist without the beings that create them No of course not, but that's not the question. The question is whether the concepts were created by the beings. In a system with 7 stars the number of stars doesn't change when all humans die or humans never even developed.
    - crazygringo 1y ago
      If humans never even developed, many philosophers would say the concepts of "number" and "7" and "star" would not exist. Others would say they all exist independently of humans. Some think abstract mathematical concepts are more privileged than physical categories like stars. That is the question. I can't even tell if you're arguing one side or the other because you say "of course not" followed by "doesn't change" when those appear to be contradictory positions. But of course there are a lot of subtleties here. There's no "of course" to any of it. These things take entire books to argue one side and try to refute the other.
      - 1718627440 1y ago
        > many philosophers If only this was the stuff humans wasted their energy on not shooting explosives to improve their ego. :-) > If humans never even developed, many philosophers would say the concepts of "number" and "7" and "star" would not exist. Yes, I'm in the camp that thinks this is just plain wrong. > abstract mathematical concepts are more privileged than physical categories like stars. I agree to that. > I can't even tell if you're arguing one side or the other Sorry, if I was unclear. You wrote: > Do or can concepts and categories exist without the beings that create them? This takes it a priori, that the categories are created by the beings. If this is true, then I think by definition this categories can't precede the beings. That's why I wrote "of course not". When things are created by beings, they don't precede them, when they are not, they do. What I already disagree with is what you already implied as a given. Honestly I don't even know how to argue for that, because this is what I think is part of the definition of the concepts and categories. When I coin some term it always happens in reaction to things I perceived with my senses. So given my senses don't lie to me, of course this is outside of me. There are also things that are in fact invented by humans, for example color names. This doesn't mean that colors don't exist independently, but the exact boundary is arbitrary. There are also concepts where I think they can be both. For example beauty. When applied to skin or shape, these are just invented, but the beauty of complexity or completeness exists outside of us. And beauty in general only exists as it points as to a thing that exists independently of us.
- EthanHeilman 1y ago
  > Isn't this article conflating our formalism of a given abstract entity (like real numbers or integers) with the abstract entity itself? It is arguing that the integers separate from the reals is the formalism and that the abstract entity is the reals. We also have a formalism of the reals, but it is closer to the abstract entity. > 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! As we create better more useful formalisms, we interact more with the formalism than the thing itself. It is like putting on oven mitts to pick up a hot tray from the oven. This has already happened with the reals! Consider the question of if the reals are well-ordered or not. The question applies to the actual entity of the reals itself, but the absoluteness theorem shows that the question has no arithmetic consequences. You can simple ignore the question. Thus, those seeking the mental comfort and utility of the formalism do not have to concern themselves with the true nature of the real number line.
  - griffzhowl 1y ago
    > It is arguing that the integers separate from the reals is the formalism and that the abstract entity is the reals. Why believe in abstract entities at all, as something distinct from the formalism? We have various formal or abstract concepts, that are useful in science and its applications, but they make contact with reality only through these uses: the natural numbers are used in counting, which is useful because there are many physical objects or events that are similar enough to each other that their differences can be neglected, then it's meaningful to count them and come up with a number for the collection, e.g. apples in a basket, planets in the solar system, or whatever. Real numbers developed from our perception of continuous physical magnitudes, and from the usefulness of applying the concept of number to these magnitudes. Then the formalism of the real numbers was developed based on mathematical constructions from the natural numbers/integers. We don't have to posit some abstract entity that the concept of real numbers refers to: it's a symbolic or mathematical construction that helps us in reasoning about (what we perceive as) continuous magnitudes, which aren't abstract, but concrete aspects of our experience of the world
    - 1y ago
- lotharcable 1y ago
  Modern science is derived from Christian Scholasticism from the middle ages so this way of talking and thinking about science as being divinely originated is only unusual in the past couple centuries or so. It is from that era that they developed systems of rigorous debate, formal logic, and things like peered reviewed papers that we call "the scientific method". As far as the history of these sorts of mathematical discussions the concept of negative numbers didn't exist until the 15 century. I am sure that each new concept was faced with some resistance and debate on its true nature before it became widely accepted. So I am sure that somebody looking through the historical record could find all sorts of wild quotes from different theologians trying to grasp new concepts and reconcile them with existing mathmatical standards.
  - griffzhowl 1y ago
    > Modern science is derived from Christian Scholasticism from the middle ages No, I don't think so. It seems much more based on ancient Greek geometry and logic, the Indian numeral system and Arabic algebra. Modern science really took off after Galileo, at the time when the ancient Greek works were recovered in Europe and could be synthesized with the arithmetic and algebra of these other cultures. Galileo himself credits the "divus" Archimedes as his main inspiration. What aspects of Christian scholasticism do you think developed into modern science?
- andrewla 1y ago
  > a quantity of stars 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?
bell-cot 1y ago
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) https://en.wikipedia.org/wiki/Closure_(topology) ) of the Rationals - but doing that was the work of man.
- Strilanc 1y ago
  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.
aaroninsf 1y ago
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.
- EthanHeilman 1y ago
  > 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 Author here. I'd describe myself as atheist/agnostic. I just dislike the God of the gaps argument. I understand its utility in debating the dishonest moving goal post arguments of young earth creationists, but taken outside of that debate, I don't think it holds water. > 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. I take religion and religious questions seriously. If I took religion less seriously I'd be religious because I enjoy religion. We owe each other a certain level of honesty on truly serious matters even if it is uncomfortable. > 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. That is certainly how a 16th Century religious Italian would understand it and I find that an interesting perspective to contrast with my own somewhat blander late-modernist beliefs. One of the reasons I enjoy reading books from prior ages is seeing how much that was taken for granted as a universal truth of that culture has changed.
jfengel 1y ago
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.
- joewferrara 1y ago
  I think you mean x*x = -1, for which I agree with your point.
  - jfengel 1y ago
    Whoops. Yep, thanks.
- tshaddox 1y ago
  I'm more troubled by the fact that almost all real numbers are uncomputable (same goes for complex numbers, of course). It's very straightforward to see that this is the case, but the mathematics involved to even begin to ponder questions like "under which operations is the set of computable reals not closed" seem to be far over my head.
  - xscott 1y ago
    Are there any operations you can even perform on the computables in the general case? Take addition, it seems simple until you try to add two computable numbers: 0.59999999999999999999999... + 0.00000000000000000000000... ------------------------------ 0.? Until you see a non-nine in that first number, or a non-zero in the second, you can't even emit the first digit of the output. From outside the black box, you don't know if the nines and zeros will stop or continue forever. I think you can make pathological cases for every arithmetic operation, so maybe (I'm not sure) none of the operations are computable. (Need to be careful with the definitions though, and I'm being pretty sloppy)
discoutdynamite 1y ago
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.
- EthanHeilman 1y ago
  I was thinking of making a "the other Agrippa" reference since with half my friends if you say Agrippa will think Camillo and the other half will think Henrichus Cornelius. I've only read Camillo Agrippa, I've been meaning to read Henrichus. Is he worth reading? What do you recommend? > 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. That does sound like the computability idea of numbers being programs. ``` In the beginning was the Word, and the Word was with God, and the Word was God. The same was in the beginning with God. All things were made by him; and without him was not any thing made that was made. ``` The "Word" in Greek is translated from logos which also means logic, rational, metric. ``` In the beginning was the Number, and the Number was with God, and the Number was God. ``` Looking around on the internet, there is a lot here: https://en.wikipedia.org/wiki/Rationes_seminales https://en.wikipedia.org/wiki/Rationes_seminales
- colanderman 1y ago
  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. [1] https://en.wikipedia.org/wiki/Definable_real_number https://en.wikipedia.org/wiki/Definable_real_number
Nevermark 1y ago
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.
- threatofrain 1y ago
  But you already pointed out the constructible reals, yes that's a a hurtful restriction for analysis, but they're still actually reals. In other words, the reals naturally arises from the discrete.
  - Nevermark 1y ago
    Integers are reals. But you can't claim something about reals because integers have that property. Constructible reals are also a subclass of reals, but you can't claim anything about the class of reals, which are vastly dominated by un-constructible reals, because constructible reals have a property. There are many reasons to doubt un-constructible numbers exist in nature. Just for starters, you can never actually define a specific un-constructible real. If you did, you would have defined it, making it constructible. An un-constructible real requires infinite information to define. Not infinite digits (pi is constructible, e is constructible), but an infinite list of uncompressible digits, or some other expression with infinite numbers of symbols! The name "reals" is highly deceptive/unfortunate. (What could be more reasonable than a "real" number?) We need a pithier name for constructible numbers, and that is what should be introduced along with algebra, calculus, trig, diff eq, etc. None of those subjects, or any practical math, ever needed the class of real numbers. The early misleading unnecessary and half-assed introduction of "reals" is an historical educational terminological aberration. (It would be nice to rename "real" numbers, to mean actually real numbers that we could actually use. But given the generations of confusion that would incur, I propose "actual numbers", to be all constructible numbers. Nobody but mathematicians, who play abstract games with higher order infinities, need "real" numbers.)
paulolc 1y ago
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.
- EthanHeilman 1y ago
  Most mathematicians, but not all, take a philosophical position of platonic idealism [0] in which numbers and other mathematical entities have an independent reality and are not invented but discovered. > All abstract concepts brought about solely by humans. That is a defendable philosophical position to hold but many thinkers disagree with it or at least admit there is more nuance [1]. People have been debating the question of the independent reality of mathematical concepts for thousands of years. If abstract concepts are brought about solely by humans, does that mean the first human to invent a proof of some property actually decides the reality of that property? If Godel hadn't proved incompleteness, could another mathematician created a completeness proof instead? On the hand, if that isn't the case and the truth or falsity of a math statement exists prior a human thinking it, doesn't it have some reality beyond human thought? What about mathematical statements which are true which can't be proven? [0]: Platonism in the Philosophy of Mathematics https://plato.stanford.edu/entries/platonism-mathematics/ https://plato.stanford.edu/entries/platonism-mathematics/ [1]: Mathematics: Discovery or Invention? https://royalinstitutephilosophy.org/article/mathematics-discovery-or-invention/ https://royalinstitutephilosophy.org/article/mathematics-dis...
- tim333 1y ago
  Things like the value of pi being 3.14159... just are what they are independent of what Sapiens want and indeed the Indiana Pi Bill of the 1897 which declared it to be 3.2 failed to change the nature of reality. That illustrates there are aspects of maths that are beyond humans just inventing them. That said I don't think there's much to be gained from dragging God into it all. Even He would probably have a headache changing pi to 3.2.
r721 1y ago
Another discussion: https://news.ycombinator.com/item?id=45065425 https://news.ycombinator.com/item?id=45065425

31 points by EthanHeilman 1y ago

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!

john-h-k 1y ago
> 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! Everything you can express in integers you can express in reals, but there are many things expressable in reals not possible in integers. It would be surprising if the formalism for a thing that completely supersets another thing had an equally simple formalism
- orlp 1y ago
  Actually in math it's very common for the more general system to be simpler. Compare for example the prime numbers with the integers, or general groups with finite simple groups and the monster group.
- tshaddox 1y ago
  I'm not suggesting the two formalisms should be equally simple. But surely it's not controversial to claim that formalizing the reals involves much more advanced mathematics (and runs into much deeper problems) than formalizing the integers. I'd argue that this disparity is slightly surprising, given that both the integers and the reals are ubiquitous in essentially all branches and levels of mathematics.
- xscott 1y ago
  > [...] but there are many things expressible in reals not possible in integers Are you sure there is anything we can express in the Reals that isn't an integer in disguise? The first answer might be the sqrt(2) or pi, but we can write a finite program to spit out digits of those forever (assuming a Turing machine with integer positions on a countably infinite length tape). The binary encoding of the program represents the number, and it only needs to be finite, not even an integer at infinity. Then you might say Chaitin's constant, but that's just a name for one value we don't know and can't figure out. You can approximate it to some number of digits, but that doesn't seem good enough to express it. You can prove a program can't emit all the digits indefinitely. And even if you could, is giving one Real number a name enough? Names are countable, and again arguably finite. It seems to me we can prove there are more Reals than there are Integer or Computable numbers, but we can't "express" more than a finite number of those which aren't computable. Integers in disguise.
  - tshaddox 1y ago
    It seems like you're suggesting that mathematicians replace the reals with the computables. This is a reasonable thing to try, and is likely of particular interest to constructivists. There's even this whole field: https://en.wikipedia.org/wiki/Computable_analysis https://en.wikipedia.org/wiki/Computable_analysis
    - xscott 1y ago
      > It seems like you're suggesting that mathematicians replace the reals with the computables. No, not any more. :-) I liked the idea a year or two ago, but I've come to believe that even the Integers are too bizarre to worry about. For now, I'm content just playing with fixed and floating point, maybe with arbitrary precision. Stuff I can reason about. I just think people are a little too casual thinking they are really using the Reals. It might be like Feynman's quote about saying you understand QM.
crazygringo 1y ago
I mean, this is one of the deepest questions of philosophy. Do or can concepts and categories exist without the beings that create them? Does tuna casserole exist independently of humans? If not, how is the idea of the number 7 different from the idea of a tuna casserole? Or what about the concept of decision by majority, which isn't as basic as 7, but doesn't have the physicality of a casserole?
- 1718627440 1y ago
  > Do or can concepts and categories exist without the beings that create them No of course not, but that's not the question. The question is whether the concepts were created by the beings. In a system with 7 stars the number of stars doesn't change when all humans die or humans never even developed.
  - crazygringo 1y ago
    If humans never even developed, many philosophers would say the concepts of "number" and "7" and "star" would not exist. Others would say they all exist independently of humans. Some think abstract mathematical concepts are more privileged than physical categories like stars. That is the question. I can't even tell if you're arguing one side or the other because you say "of course not" followed by "doesn't change" when those appear to be contradictory positions. But of course there are a lot of subtleties here. There's no "of course" to any of it. These things take entire books to argue one side and try to refute the other.
    - 1718627440 1y ago
      > many philosophers If only this was the stuff humans wasted their energy on not shooting explosives to improve their ego. :-) > If humans never even developed, many philosophers would say the concepts of "number" and "7" and "star" would not exist. Yes, I'm in the camp that thinks this is just plain wrong. > abstract mathematical concepts are more privileged than physical categories like stars. I agree to that. > I can't even tell if you're arguing one side or the other Sorry, if I was unclear. You wrote: > Do or can concepts and categories exist without the beings that create them? This takes it a priori, that the categories are created by the beings. If this is true, then I think by definition this categories can't precede the beings. That's why I wrote "of course not". When things are created by beings, they don't precede them, when they are not, they do. What I already disagree with is what you already implied as a given. Honestly I don't even know how to argue for that, because this is what I think is part of the definition of the concepts and categories. When I coin some term it always happens in reaction to things I perceived with my senses. So given my senses don't lie to me, of course this is outside of me. There are also things that are in fact invented by humans, for example color names. This doesn't mean that colors don't exist independently, but the exact boundary is arbitrary. There are also concepts where I think they can be both. For example beauty. When applied to skin or shape, these are just invented, but the beauty of complexity or completeness exists outside of us. And beauty in general only exists as it points as to a thing that exists independently of us.
EthanHeilman 1y ago
> Isn't this article conflating our formalism of a given abstract entity (like real numbers or integers) with the abstract entity itself? It is arguing that the integers separate from the reals is the formalism and that the abstract entity is the reals. We also have a formalism of the reals, but it is closer to the abstract entity. > 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! As we create better more useful formalisms, we interact more with the formalism than the thing itself. It is like putting on oven mitts to pick up a hot tray from the oven. This has already happened with the reals! Consider the question of if the reals are well-ordered or not. The question applies to the actual entity of the reals itself, but the absoluteness theorem shows that the question has no arithmetic consequences. You can simple ignore the question. Thus, those seeking the mental comfort and utility of the formalism do not have to concern themselves with the true nature of the real number line.
- griffzhowl 1y ago
  > It is arguing that the integers separate from the reals is the formalism and that the abstract entity is the reals. Why believe in abstract entities at all, as something distinct from the formalism? We have various formal or abstract concepts, that are useful in science and its applications, but they make contact with reality only through these uses: the natural numbers are used in counting, which is useful because there are many physical objects or events that are similar enough to each other that their differences can be neglected, then it's meaningful to count them and come up with a number for the collection, e.g. apples in a basket, planets in the solar system, or whatever. Real numbers developed from our perception of continuous physical magnitudes, and from the usefulness of applying the concept of number to these magnitudes. Then the formalism of the real numbers was developed based on mathematical constructions from the natural numbers/integers. We don't have to posit some abstract entity that the concept of real numbers refers to: it's a symbolic or mathematical construction that helps us in reasoning about (what we perceive as) continuous magnitudes, which aren't abstract, but concrete aspects of our experience of the world
  - 1y ago
lotharcable 1y ago
Modern science is derived from Christian Scholasticism from the middle ages so this way of talking and thinking about science as being divinely originated is only unusual in the past couple centuries or so. It is from that era that they developed systems of rigorous debate, formal logic, and things like peered reviewed papers that we call "the scientific method". As far as the history of these sorts of mathematical discussions the concept of negative numbers didn't exist until the 15 century. I am sure that each new concept was faced with some resistance and debate on its true nature before it became widely accepted. So I am sure that somebody looking through the historical record could find all sorts of wild quotes from different theologians trying to grasp new concepts and reconcile them with existing mathmatical standards.
- griffzhowl 1y ago
  > Modern science is derived from Christian Scholasticism from the middle ages No, I don't think so. It seems much more based on ancient Greek geometry and logic, the Indian numeral system and Arabic algebra. Modern science really took off after Galileo, at the time when the ancient Greek works were recovered in Europe and could be synthesized with the arithmetic and algebra of these other cultures. Galileo himself credits the "divus" Archimedes as his main inspiration. What aspects of Christian scholasticism do you think developed into modern science?
andrewla 1y ago
> a quantity of stars 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?

bell-cot 1y ago

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) https://en.wikipedia.org/wiki/Closure_(topology) ) of the Rationals - but doing that was the work of man.

Strilanc 1y ago
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.

aaroninsf 1y ago

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.

EthanHeilman 1y ago
> 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 Author here. I'd describe myself as atheist/agnostic. I just dislike the God of the gaps argument. I understand its utility in debating the dishonest moving goal post arguments of young earth creationists, but taken outside of that debate, I don't think it holds water. > 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. I take religion and religious questions seriously. If I took religion less seriously I'd be religious because I enjoy religion. We owe each other a certain level of honesty on truly serious matters even if it is uncomfortable. > 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. That is certainly how a 16th Century religious Italian would understand it and I find that an interesting perspective to contrast with my own somewhat blander late-modernist beliefs. One of the reasons I enjoy reading books from prior ages is seeing how much that was taken for granted as a universal truth of that culture has changed.

jfengel 1y ago

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.

joewferrara 1y ago
I think you mean x*x = -1, for which I agree with your point.
- jfengel 1y ago
  Whoops. Yep, thanks.
tshaddox 1y ago
I'm more troubled by the fact that almost all real numbers are uncomputable (same goes for complex numbers, of course). It's very straightforward to see that this is the case, but the mathematics involved to even begin to ponder questions like "under which operations is the set of computable reals not closed" seem to be far over my head.
- xscott 1y ago
  Are there any operations you can even perform on the computables in the general case? Take addition, it seems simple until you try to add two computable numbers: 0.59999999999999999999999... + 0.00000000000000000000000... ------------------------------ 0.? Until you see a non-nine in that first number, or a non-zero in the second, you can't even emit the first digit of the output. From outside the black box, you don't know if the nines and zeros will stop or continue forever. I think you can make pathological cases for every arithmetic operation, so maybe (I'm not sure) none of the operations are computable. (Need to be careful with the definitions though, and I'm being pretty sloppy)

discoutdynamite 1y ago

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.

EthanHeilman 1y ago
I was thinking of making a "the other Agrippa" reference since with half my friends if you say Agrippa will think Camillo and the other half will think Henrichus Cornelius. I've only read Camillo Agrippa, I've been meaning to read Henrichus. Is he worth reading? What do you recommend? > 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. That does sound like the computability idea of numbers being programs. ``` In the beginning was the Word, and the Word was with God, and the Word was God. The same was in the beginning with God. All things were made by him; and without him was not any thing made that was made. ``` The "Word" in Greek is translated from logos which also means logic, rational, metric. ``` In the beginning was the Number, and the Number was with God, and the Number was God. ``` Looking around on the internet, there is a lot here: https://en.wikipedia.org/wiki/Rationes_seminales https://en.wikipedia.org/wiki/Rationes_seminales
colanderman 1y ago
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. [1] https://en.wikipedia.org/wiki/Definable_real_number https://en.wikipedia.org/wiki/Definable_real_number

Nevermark 1y ago

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.

threatofrain 1y ago
But you already pointed out the constructible reals, yes that's a a hurtful restriction for analysis, but they're still actually reals. In other words, the reals naturally arises from the discrete.
- Nevermark 1y ago
  Integers are reals. But you can't claim something about reals because integers have that property. Constructible reals are also a subclass of reals, but you can't claim anything about the class of reals, which are vastly dominated by un-constructible reals, because constructible reals have a property. There are many reasons to doubt un-constructible numbers exist in nature. Just for starters, you can never actually define a specific un-constructible real. If you did, you would have defined it, making it constructible. An un-constructible real requires infinite information to define. Not infinite digits (pi is constructible, e is constructible), but an infinite list of uncompressible digits, or some other expression with infinite numbers of symbols! The name "reals" is highly deceptive/unfortunate. (What could be more reasonable than a "real" number?) We need a pithier name for constructible numbers, and that is what should be introduced along with algebra, calculus, trig, diff eq, etc. None of those subjects, or any practical math, ever needed the class of real numbers. The early misleading unnecessary and half-assed introduction of "reals" is an historical educational terminological aberration. (It would be nice to rename "real" numbers, to mean actually real numbers that we could actually use. But given the generations of confusion that would incur, I propose "actual numbers", to be all constructible numbers. Nobody but mathematicians, who play abstract games with higher order infinities, need "real" numbers.)

paulolc 1y ago

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.

EthanHeilman 1y ago
Most mathematicians, but not all, take a philosophical position of platonic idealism [0] in which numbers and other mathematical entities have an independent reality and are not invented but discovered. > All abstract concepts brought about solely by humans. That is a defendable philosophical position to hold but many thinkers disagree with it or at least admit there is more nuance [1]. People have been debating the question of the independent reality of mathematical concepts for thousands of years. If abstract concepts are brought about solely by humans, does that mean the first human to invent a proof of some property actually decides the reality of that property? If Godel hadn't proved incompleteness, could another mathematician created a completeness proof instead? On the hand, if that isn't the case and the truth or falsity of a math statement exists prior a human thinking it, doesn't it have some reality beyond human thought? What about mathematical statements which are true which can't be proven? [0]: Platonism in the Philosophy of Mathematics https://plato.stanford.edu/entries/platonism-mathematics/ https://plato.stanford.edu/entries/platonism-mathematics/ [1]: Mathematics: Discovery or Invention? https://royalinstitutephilosophy.org/article/mathematics-discovery-or-invention/ https://royalinstitutephilosophy.org/article/mathematics-dis...
tim333 1y ago
Things like the value of pi being 3.14159... just are what they are independent of what Sapiens want and indeed the Indiana Pi Bill of the 1897 which declared it to be 3.2 failed to change the nature of reality. That illustrates there are aspects of maths that are beyond humans just inventing them. That said I don't think there's much to be gained from dragging God into it all. Even He would probably have a headache changing pi to 3.2.

r721 1y ago

Another discussion: https://news.ycombinator.com/item?id=45065425 https://news.ycombinator.com/item?id=45065425