"The Computer and the Brain" by John von Neumann

[dmm]

This is the last (and somewhat unfinished) short book by John von Neumann.

It's a short book, the PDF file is only 97 pages, the download link is visible from here: scholar.google.com/scholar?q=%22The+computer+and+the+brain%22"

I've read this very interesting book during the last few days, and I'll record my impressions in the comments to this post.

Comments

[dmm]

Then he is making an observation that neural computations have to be robust to high levels of noise and to be able to work well at reduced precision.

Of course, this observation can now be considered as a correct prediction for artificial neural nets as well.

He conjectures that the "intrinsic math of neural machinery in the brain" is, therefore, quite different from our "conventional math", but does not elaborate.

I am going to elaborate on what this "intrinsic math" might be further in this thread.

[dmm]

So, on the surface of it, it seems that "changes to obtain 'intrinsic' math" satisfying von Neumann's desiderata are quite minimal. One needs to avoid noise amplification, so one needs to make sure to use non-expansive maps/to make sure that derivatives not exceed 1, and that's more or less all - just some moderate constraints here, not radical changes.

In addition, one should remember that recurrent machines X_{n+1}=F(X_n) work reasonable well, when F is close to the identity map, e.g. https://dmm.dreamwidth.org/19100.html .

Cf. also the comment on "The Analog Procedure" above.

***

At the same time, if one want to explore the math, specifically oriented towards probability and statistics and/or interval numbers, and then perhaps to add the ability to work with partial contradictions in probabilistic and interval setups (https://www.cs.brandeis.edu/~bukatin/dmm-probabilistic-samples.pdf ; https://www.cs.brandeis.edu/~bukatin/PartiallyInconsistentIntervalNumbers.pdf), this is a fertile ground for such experiments.

And then one might try to formulate all this in topos terms, just like people explored this for quantum theory, e.g. https://arxiv.org/abs/1107.1083 "Unsharp Values, Domains and Topoi".

So, one could still use von Neumann remarks on the need for different "intrinsic math" for neural computations as an inspiration for various non-trivial explorations here, even if rather mild changes are quite sufficient to satisfy von Neumann's desiderata from a superficial viewpoint.

Perhaps, there are reasons to move beyond this superficial viewpoint, even if they are not sufficiently articulated in the von Neumann's text.

[dmm]

One more thing: von Neumann writes that neurons tend fire periodically or near periodically, at least when the stimulation is really strong (because of the absolute refractory period), and, perhaps, even with moderately strong stimulation when they are in the relative refractory period ("Nature of the System of Notations Employed: Not
Digital but Statistical" section, page 88-89).

I am always interesting in the attempts to move from rate coding to spike synchronizations and oscillations associated with those synchronizations. However, I usually assumed that non-synchronized neurons yield something close to Poisson spike trains. Regardless, of what it is in actual biology, it might be quite fruitful to take a hint from von Neumann and consider periodic behavior even for non-synchronized neurons, and proceed with synchronization models from that basis.

[dmm]

This is the last comment I promised above. The rather tragic history of von Neumann writing the material which became this book while dying from cancer is described in detail in the preface.

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I learned about this book here

https://twitter.com/JohnMeuser/status/1293339704226656258

and then John pushed me to actually read it

https://twitter.com/JohnMeuser/status/1296918998391697415

and there are various discussions between us around those threads, e.g.

https://twitter.com/JohnMeuser/status/1297639656922767362

(He is obviously seeing something else there, not what I am seeing, e.g.

https://twitter.com/JohnMeuser/status/1297640700872450050

https://twitter.com/JohnMeuser/status/1297641242474446848

https://twitter.com/JohnMeuser/status/1297639967280291844

Most of what I wrote in those twitter threads is consolidated in my comments to this post.)