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I couldn't decide whether this question is more appropriate to post here or at the philosophy SE, but I thought I'd give people with a mathematical perspective the opportunity to help me decide. I'm happy to migrate the question if people think it necessary.

Anyway, the question is about a famous quote in von Neumann's The Computer and the Brain (published posthumously), that goes:

When we talk mathematics, we may be discussing a secondary language, built on the primary language truly used by the central nervous system.

Also

Thus logics and mathematics in the central nervous system, when viewed as languages, must structurally be essentially different from those languages to which our common experience refers.

I've read the relevant passages a few times, and far as I can see he is trying to convey a statement along the lines of: "The mathematical or logical langauge of the nervous system is not the same as that we use when we do/talk about mathematics."

I can't decide whether the quote is vacuous or profound. From a modern perspective, we may say that the nervous system is a biological neural network, with (as von Neumann himself suggested in this book) low precision but high reliability. And then, presumably, this neural net is Turing-complete, and the actual way in which computations are performed is not important - they are still computations, and they could just as well have been run on some sort of a cellular automaton.

So my question is: are there any concrete consequences of von Neumann's observation within the realm of metamathematics? For instance, is there some theorem that becomes relevant? If the answer is 'no, this is mathematically irrelevant' then I suppose this is a philosophical question, in which case I'd be interested in implications for the philosophy of mathematics.

Level of my question: I am somewhat familiar with various notions that may be relevant, like first and second order logic, the Entscheidung problem, Gödel's theorems etc, but I was trained as a physicist, not a mathematician.

TLDR: was von Neumann onto something fundamental here about the nature of mathematics as a language , or is it merely a subtlety around the physical instantiation of systems that compute, and their architecture?

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    $\begingroup$ I think the statements are likely formally vacuous, but also metaphysically profound. I think your questions are open to interpretation so probably not strictly appropriate to this site, however they are well formed, and I think it is good to have non philosopher mathematicians' thoughts on such topics recorded. Personally, in a strict sense, I would say that the language of the brain is neither logical nor mathematical. I think the functioning of the brain is probably more like natural language than we might assume, meaning that much ambiguity is present and it is prone to errors. $\endgroup$ – jdods Dec 1 '17 at 13:44
  • $\begingroup$ I find the cellular autonoma theory or neural network interpretation of the brain rather implausible. People are rather terrible at these type of tasks and assuming the brain is capable of them seems suspect. Indeed only a rare brain, properly conditioned can even come close to approximating them. It's too fallible and intimately connected to it's environment to be described by deterministic rules. $\endgroup$ – CyclotomicField Dec 1 '17 at 13:46
  • $\begingroup$ @CyclotomicField The statement is not that humans can explicitly perform the types of calculations that neural nets can, but that our information processing ability is captured by a Turing machine model of computation, and that neural nets are Turing-complete. I believe this is currently the mainstream position, although there are certainly alternative views. $\endgroup$ – Martin C. Dec 1 '17 at 14:29
  • $\begingroup$ @MartinC. That's the part I find so implausible. At best these analogies are toy problems to attempt and simply the brain but they are not accurate nor are they well motivated by empirical evidence rather they are just guesses that are computationally feasible. Speculative at best. $\endgroup$ – CyclotomicField Dec 1 '17 at 16:35
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I would say it's somewhere in between what you say at the end:

Presumably von Neumann viewed the 'language of the brain', i.e. the way that the brain represents information, and the processes defined over those representations in order to 'infer' or 'compute', as significantly different than the 'language of humans', i.e. the languages that mathematicians, physicists and, for that matter, all of us using natural language are using. Think, for example, of the way artifical neural networks represent and process information versus more traditional symbolic languages and processes.

As such, characterizing the difference as 'mere' differences in physical implementation I think falls a bit short of the point that Von Neumann was trying to make: in Von Neumann's eyes we would find nothing like the kinds of algorithms that mathematicians typically employ in the brain, even if we do go beyond implementational detail: the brain's computations presumably are really quite different in nature. When we catch a ball, for example, we shouldn't think that our brains are solving that problem by describing it using certain kinds of Newtonian mechanical/dynamical equations and solving those using the typical methods we as mathematicians employ.

But yes, at the same time, Von Neumann would probably agree that whatever the brain is doing is still within the realm of Turing-computability. It's just that this realm is so vast, that trying to draw any parallels between the 'language of the brain' and the 'language of mathematicians' is pretty futile.

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  • $\begingroup$ I like your example of catching a ball. I take it the short version of your answer is 'no, this has no formal mathematical implications'? $\endgroup$ – Martin C. Dec 1 '17 at 14:24
  • $\begingroup$ The brain has previous experience catching the ball. So it sees the current experience of catching the ball as being similar to those past experiences (having similar patterns of neuron firings), therefor it implements a similar reaction as it has been conditioned (learned) to do so. That's just my speculation. Of course, the brain is also very good at improvising solutions, but again, I think this can be related to the idea of comparing the patterns of neuron firings, and just coming up with a response (pattern of neuron firings) according to some averaging of previous conditioned responses. $\endgroup$ – jdods Dec 1 '17 at 17:10
  • $\begingroup$ @MartinC. Right, nothing formal. But, I have often wondered if in order to get a handle on translating the complexitiy of the brain into something understandable could make use of, if not requires, some new mathematical tools. $\endgroup$ – Bram28 Dec 1 '17 at 17:50
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    $\begingroup$ @jdods Yeah, that would be my guess as well: mostly learned patterns, but patterns that are invariant over all kinds of transformations, so that color, size, texture, etc, etc. don't matter. I also like the idea of the brain using a percptuo-motor feedback cycle: just keep looking at the ball, and if in your field of vision it goes left, then you move left; if it moves right, you move right. How far? How long? Doesn't matter! Just keep your eyes on the ball follow this super simple recipe, and you should be in the ball's path. No calculations needed whatsoever! $\endgroup$ – Bram28 Dec 1 '17 at 17:55

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