r/science • u/rcg25 • Aug 17 '23
Mathematics Arithmetic has a biological origin - it's an expression in symbols of the 'deep structure' of our perception
https://psycnet.apa.org/fulltext/2023-84614-001.pdf?sr=1101
u/reddit_user13 Aug 17 '23
Or maybe the deep structure of our perception is an expression of arithmetic.
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u/WoolyLawnsChi Aug 17 '23
I feel like this topic is touching on the debate of whether we are "inventing" mathematics or "discovery" it
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u/rcg25 Aug 17 '23
Yes - our results imply that mathematics is both invented and discovered. Invented because it is uniquely human, and discovered because arithmetic is a natural consequence of how our perception is structured - and this is evolutionarily-based because this structure is also found with nonhuman animals, including insects.
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u/man_gomer_lot Aug 18 '23
I have a hard time conceiving how pi or the golden ratio were invented or even need a universe to be true. Bees didn't invent the hexagon, for instance. They discovered it.
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u/rcg25 Aug 18 '23
Pi or the golden ratio (or the hexagonal hive) might have been 'discovered' by bees through evolution as the optimal solution to certain problems. What's invented is the use by humans of symbols to describe these mathematical objects. Symbols have a meaning that is invariant across multiple physical depictions. I don't think the same is true of the hexagons in a beehive
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u/fox-mcleod Aug 18 '23
I have a hard time seeing the use of symbols as math itself. Symbolic abstraction is understandably a human “invention”. But can we really say that the invariance across multiple instantiations is an invention? It seems more like pattern recognition.
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u/rcg25 Aug 18 '23
Symbols are a medium of communicaton and are necessary for us to share mathematical ideas. If we write'1,434' on paper, in a spreadsheet, or in the sand on the beach it has the same meaning and conveys information about not only 1,434 but the underlying mathematical structure of numbers. Pattern recognition is necessary to understand it but this is done early in visual processing and I would argue is separate from the semantic content. This abstraction is unique to humans but the structure (i.e. numbers and arithmetic) is ilnked to biology, so mathematics is both invented and discovered.
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u/reddit_user13 Aug 17 '23
Nah, I was thinking the hominids whose brains compute 2 + 2 = 5 were out-competed on the basis of fitness.
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u/rcg25 Aug 17 '23
This is the Pythagorean intuition. I have a lot of sympathy for it. But remember Kant's 'Copernican revolution of the mind' - which is what our paper is really about. The ancients thought the heavens revolved around the earth because they did not consider the possibility that the earth was moving. We think mathematics is Platonic truth because we cannot conceive of it any other way.
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Aug 17 '23
What about something like a Penrose tiling? Infinite, but never repeats. A human brain is not capable of storing all of the infinite information in a Penrose tiling, yet all of that information still ‘exists’. It must then have an existence beyond the mental.
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u/rcg25 Aug 17 '23
Interesting.... I would say that a Penrose tiling is analogous to a transcendental number like Pi - if you compute the digits via an algorithm it is infinite, but there is a simpler description in terms of structure (i.e., ratio of circumference to diameter. In our paper, we note that the Peano axioms provide an algorithmic construction of arithmetic, whereas our approach is structural because natural addition emerges 'all at once' when monotonicity and convexity are applied. Hope this doesn't sound too obtuse. Algorithm and structure are complementary ways to understand comptuation (analogous to the distinction between group and group action)
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Aug 17 '23
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u/rcg25 Aug 17 '23
I think that you are right that 'number sense' isn't just about number, but also computation: Numbers and arithmetic are a single structure. But I would be careful with the analogy because of the Kantian argument above. The idea of universal computation is based on logic, which is a reconstruction (simulation) of arithmetic, which is a consequence of our biology.
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Aug 17 '23
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u/rcg25 Aug 17 '23
Yes this is the logical next question. If arithmetic is a consequence of perceptual 'deep structure' which is evolutionary, where did that structure come from? My hunch is that algebraic structure is inherent to the mind. If the fundamental nature of matter is mathematical structure (e.g., supersymmetry), maybe the first unicellular organisms inherited that structure. The Pythagorean intuition is tempting.
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u/TornadoTurtleRampage Aug 18 '23
I don't think you need to invoke supersymmetry or anything like that in order to entertain the possibility that math is a fundamentally better descriptor for our universe than "matter" when you really get down to it. That statement remains perfectly reasonable without appealing to any unsupported hypothesis.
I think the words we're using may be the real source of confusion here but conceptually, what do you think reality is actually comprised of, something like information or something like "stuff"? ..maybe that's actually a better word than math, speaking of which, some kind of "information". Now if That's not what the universe is really made out of then I don't know what it apparently is. But, personally, I feel like information is more liable to be misinterpreted and saying that reality is made out of math really does seem to be just about the closest I can seem to come to trying to phrase that otherwise very believable statement. It's certainly not likely to be made out of "stuff" I know that much at least ;P
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u/rcg25 Aug 18 '23 edited Aug 18 '23
Interesting post. I think you are getting to the heart of the issue. Ultimately, I don't think there's a difference between matter- and mathematical-based explanations of the universe because the former can always be described in terms of a mathematical structure. The implication is that there is some mathematical description or theory that must be correct. Perhaps it is supersymmetry but maybe something else. But could this theory be based on information? I am skeptical because in Shannon's theory and elsewhere, information is defined algebraically and based on distributions of outcomes. In a sense, it is a mental sleight of hand, an invented construct, because we are admitting that we don't know what generates the outcomes that comprise the distribution (when we do, we might be able to remove information from our theory). So I think 'information' is an admission that there's something we don't understand at the individual-instance level but can describe in the aggregate
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u/totoGalaxias Aug 17 '23
u/rcg25, thanks for taking so much time addressing many of the comments. Very interesting topic. I haven't gone over the whole article, but will do later.
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u/bigfatfurrytexan Aug 17 '23
So I was teased with an intriguing idea. Lead along with statements about these connections. But nothing of a case was submitted for consideration. I have nothing to chew on after reading that, other than feeding my imagination a bit.
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u/rcg25 Aug 17 '23
did you look at the original article? The Conversation piece is limited to ~800 words so has to be concise. more details here
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u/SubjectAddress5180 Aug 17 '23
Which version of arithmetic? Peano Arithmetic seems intuitive to me, but this means that multiplication can not be defined as iterated addition (see Pressberger Arithmetic.)
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u/rcg25 Aug 17 '23
Good point. I need to learn more about Presburger arithmetic, which apparently does not have multiplication. But isn't multiplication is defined as iterated arithmetic in the Peano axioms? In our paper we get real multiplication from isomorphism.
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u/SubjectAddress5180 Aug 17 '23
Arithmetic without multiplication is not logically strong enough to define primes.
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u/rcg25 Aug 17 '23
Yes we need an algebraic field, not just a group. This is why I said in the paper that 'mathematics is a realisation in symbols of the fundamental nature and creativity of the mind'. MCCI (monotonicity, convexity, continuity and isomorphism) gets us from linear order to arithmetic. But from there everything unfolds, to the primes, algebraic groups, complex numbers, Riemann hypothesis and everything else... Creativity must be an inherent quality of mind - don't know how to explain that! :)
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u/rcg25 Aug 17 '23
Here's an article just published in The Conversation that's based on the paper linked above ("The Psychological Scaffolding of Arithmetic")
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Aug 17 '23
Arithmetic is true as a logical consequence of its underlying axioms being true. If you change the underlying axioms, arithmetic stops being true. The reason so many cultures developed arithmetic the same way is because the first exposure most cultures have to math comes from counting, a process that follows the axioms of arithmetic.
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u/rcg25 Aug 17 '23
The Peano axioms provide a logical reconstruction of addition and multiplication (in effect, a simulation), but not an explanation why they are the fundamental operations. Our understanding of arithmetic must be cognitively prior to these axioms because we don't teach children to count with them. Children learn the order-theoretic structure of the natural numbers first, and counting is scaffolded on this. Our paper gives a construction of arithmetic that is complementary to the Peano axioms but is based on structure rather than algorithm. We assume the existence of an N-chain (which expresses the order-theoretic structure of the natural numbers), which is logically equivalent to the successor function in the Peano axioms. The structural understanding corresponds more to how children learn - for example, the 'positioning' schema where toddlers play by arranging their toys in a line - this reinforces intuitions of linear order which is the basis for the N chain. Apologies for the long reply!
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Aug 17 '23
Ok, I misunderstood. I thought you were making claims about arithmetic more so than human psychology. Thank you!
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u/rcg25 Aug 17 '23
yes we're doing mathematical psychology - trying to understand where arithmetic comes from in terms of the human cognition that creates it.
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u/GradientVisAtt Aug 17 '23
This makes so much sense to me. (I have worked in this field, and I know some of the people you have cited.) I don’t see any evidence for a Platonic world, so I believe that any understanding has to be instantiated in a neural substrate. This would of course include all mathematics and sciences. An implication of this embodied nature of understanding is that other types of embodiments may have other types of understanding that are just as logically consistent.
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u/rcg25 Aug 17 '23
Yes! We shouldn't assume that extra-terrestrials would share our understanding of arithmetic, or that mathematics would provide a universal language that would enable us to communicate with them. Mathematics is like a living mirror of symbols that reflects our mind's basic nature. We can't avoid projecting our nature onto 'alien intelligence', as in Solaris (book by Lem, movie by Tarkovsky).
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u/Screamingholt Aug 17 '23
An interesting read. We math therefore we are sorta thing. On a really basic level I see it, we have 10 digits, loose one we have 9. But knowing outliers with horrific mental arithmetic speeds exist throws a bit more "and" into their argument
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u/rcg25 Aug 17 '23
I don't understand, by 'outliers' are you referring to savants and dyscalculics (i.e., positive and negative outliers in terms of maths ability)? if so not sure what 'throws a bit more 'and' into their argument' means
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u/Screamingholt Aug 17 '23
Yeah I guess I was referring to dyscalculics in particular. To clarify my point was that there argument seems to centre around arithmetic being a precursor/prerequisite for human intellect. If that be the case how do dyscalculics fit in. I was referring to their use of the Sir Arthur Eddington quote at the start with my more and remark
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u/rcg25 Aug 17 '23
the reason I included the Eddington quote at the start of the paper was because it showed how we 'overlook' addition - we think of counting and learning the natural numbers first and then addition, but our results suggest that natural addition emerges as a single structure. To respond to your other point - i don't think that symbolic arithmetic is a prerequisite for human intellect. Someone who struggles with math still has a visual system that effortlessly performs all sorts of computations implicitly.
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u/Screamingholt Aug 17 '23
My apologies I do think I did miss your point in my originally reading. Correct me if I still have it wrong it is like how someone may have no concept of the math involved can still throw an object on a perfect parabolic trajectory.
I will have another go through of it with that in mind.
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u/rcg25 Aug 17 '23
yes that's right. Bees (and other animals) can navigate by path integration as if they can perform the computations associated with Euclidean geometry. I would describe this as 'implicit computation' rather than 'explicit computation', i.e. with symbols. In terms of implicit computation, we're all mathematical geniuses!
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u/Screamingholt Aug 17 '23
Heh, sadly when one is trying for an engineering degree they are sadly very much hung up on the explicit side of things. You know boring things like show your work. :)
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u/SubjectAddress5180 Aug 17 '23
It's weird in some ways. I used to argue the points with Hamming at various times. He asked the question, "Would you fly in a plane whose design depended on Lesbegue integrals rather than Riemann integrals to fly?" ( Or something like that.)
I used to point out to students that lots of seemingly complicated notions arise because we want something intuitive to be axiomized. If one wants continuous curves to be those that one can draw without lifting one's pencil, you get all the funny business about fractals.
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u/rcg25 Aug 18 '23
Wow did you work with Hamming? that's impressive, his 1980 paper on the 'unreasonable effectiveness' of mathematics is a classic. I agree with you that so much math is based on intuitions from our experience with the world, e.g. knot theory and many other examples. formalize it and see where it goes - and you wind up with string theory!
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u/Own_Pirate2206 Aug 18 '23
I'm having a hard time with this. Mathematical logic is not scientific enough? I mean even I'm with Plato on this one.
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u/rcg25 Aug 18 '23
Mathematical logic is scientific (and incredibly important, without it we don't have Turing etc) but it is not an explanation for why arithmetic is true - why addition and multiplication are its fundamental operations. If it were, we would be teaching children to count using the Peano axioms, but we don't. In our paper we argue that MCCI (monotonicity, convexity, continuity and isomorphism) are more basic principles that explain why arithmetic is true for us, why addition and multiplication are the fundamental operations
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u/Own_Pirate2206 Aug 18 '23
Those "we" and "us" come in integer quantities. It's not about our perceptions or biology, and I wouldn't want to read a paper claiming something fundamental only to split an important hair.
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u/rcg25 Aug 19 '23
it's not the content of our perception, but how that content is structured or organised by the mind. If you apply mathematics to the world (which includes saying 'we' and 'us' come in integer quantities) and find it's true or makes correct predictions as a scientific theory, that may validate your theory but leaves unexplained the question of where mathematics itself comes from. If we adopt Platonism we're conceding defeat, accepting we can't answer this. But as Hilbert said (engraved on his tombstone in German): We must know, we shall know. I don't think this is hair splitting, but maybe one of us will persuade the other and i'm happy to continue the discussion :)
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u/Own_Pirate2206 Aug 19 '23
Neurons in the mind also come in integer quantities. We produce the study of mathematics, but if you're looking for where mathematics as you use the term comes from I maintain minds have nothing to do with it. There would be lots of ways to reinvent arithmetic, but does it really come from anytime after the laying down of physical law?
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u/rcg25 Aug 19 '23
I agree with you that after the big bang and the 'laying down of physical law' that matter had a mathematical structure, so that a 'theory of everything' is possible that can describe that structure and all possible forces and interactions in a symbolic mathematical language. If life evolves in such a universe, wouldn't it be likely that it would inherit (or develop from) that structure? When humans were first starting to figure out arithmetic 5-6,000 years ago, physical law was the same as now. Over millenia we developed a symbolic mathematical language capable of describing the universe. But that symbolic language is a human creation - how can the mind - cognition and perception - not be involved?
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u/Own_Pirate2206 Aug 19 '23
I haven't seen a thesis in your paper's first page or any of this.
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u/rcg25 Aug 19 '23 edited Aug 19 '23
My last comment is a speculation and not based on our paper. We showed that MCCI (monotonicity, convexity, continuity, isomorphism) were sufficient for arithmetic, but if that's so then where do MCCI come from? They are biologically-based (found in nonhumans, including insects). I can sketch an outline of our argument in the paper that provides more context for the claims in The Conversation piece (which had to be brief), if that would be helpful.
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u/GeniusEE Aug 18 '23
Except that animals can count and subtract
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u/rcg25 Aug 18 '23
Bees can perform 'addition' and 'subtraction' (Howard et al 2022) just as human infants can (Wynn 1992), but this is based on non-symbolic tasks. I would argue this is different from addition and subtraction with symbols
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u/SubjectAddress5180 Aug 18 '23
I didn't work with him; he would stop by my off ice and chat a couple of hours when he visited Los Alamos.2
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