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Digital Mathematics and Non-digital Mathematics

Trying to understand anti-formalists

By K. Podnieks

 This work is licensed under a Creative Commons License and is copyrighted © 1999 by  me, Karlis Podnieks.

(Please, excuse me, if the following notes seem trivial to you.)

A thesis proposed by Stanislaw Lem (Polish science fiction writer) in his book "Summa Technologiae" (see the Chapter "Madness and Method"):

Mathematicians are mad tailors: they are making "all the possible clothes" hoping to make also something suitable for dressing... (sorry - my own English translation, the initial version of this aphorism may be due to David van Dantzig, see Quotations by David van Dantzig).

(Polonius says in "Hamlet" - though not about mathematics: "Though this be madness, yet there is method in't.")

Gordon Fisher: But it may be, may it not, that all possible clothes won't cover all that's physically possible?

This question forced me to perform the following chain of reasoning:

Formal theories are physical objects. I.e. applying such a theory to some natural or technical phenomenon means exploiting of a really existing (physical!) isomorphism between two physical objects - the theory and the phenomenon. But, of course, formal theories are physical objects of a specific kind - I would call them "digital" objects because they all can be implemented (by definition!) as programs of digital computers.

Note. You may wish to use the term "discrete" instead of my "digital". I like the latter more (digital computers, digital TV, digital music records, digital phones etc.).

Thus, the question could be reformulated as follows: but it may be, may it not, that all possible digital structures cover all that's physically possible? I.e., may be, to cover some physical phenomena we may need a non-digital ("non-digitalizable"!) structures as models?

Note. The "continuum" formalized, for example, in Zermelo-Fraenkel set theory ZFC, should be regarded as a "digital" structure, because digital computers can generate all theorems (of ZFC) about this "continuum".

Some problems

1. Is human brain capable of creating non-digitalizable models? Is our informal intuition of natural numbers a kind of such non-digitalizable models?

2. May be, we continuously fail to capture space/time/continuum "as a whole" because we are trying to do this by using only digital models?

3. You can achieve any level of fidelity in digital recording of music, and any level of fidelity in predicting of Solar eclipses by using the digital mechanics. But, of course, each physical phenomenon is the best possible "model" of itself! Do we need more than this? I.e. do we need non-digitalizable models at all?

4. For me as formalist, the formalizable (i.e. digitalizable) mathematics is the "only true" kind of mathematics. But, may be, the "occultist style" opponents of the formalist philosophy of mathematics simply are searching for non-digitalizable models? I.e., may be, they are normal people?

5. Perhaps, the enthusiasts of quantum computers already know about these problems for a long time?

October 4, 1999


Added October 16, 2002:
An attempt (May 12, 1999) to answer question 1 positively, by Roger Penrose - see (thanks to Valentin N. Pavlov).

Added April 25, 2003:
Hilary Putnam: "That's all well and good internally, Mr. Boolos, but in what relation does your thesis stand to the universe?"
George Boolos: "It's part of it".
Quoted after Judith Jarvis Thomson, see

Added July 11, 2003
Attempts to answer questions 2 and 3 negatively - see Rechnender Raum (1967) by Konrad Zuse, Digital Philosophy by Edward Fredkin (and A New Kind of Science by Stephen Wolfram).

Added November 19, 2003
See "Real" Analysis is a Degenerate Case of Discrete Analysis (November 26, 2001) by Doron Zeilberger:
"... the conventional wisdom, fooled by our misleading "physical intuition", is that the real world is continuous, and that discrete models are necessary evils for approximating the "real" world, due to the innate discreteness of the digital computer.
Ironically, the opposite is true. The
Continuous analysis and geometry are just degenerate approximations to the discrete world, made necessary by the very limited resources of the human intellect. While discrete analysis is conceptually simpler (and truer) than continuous analysis, technically it is (usyally) much more difficult."

Added August 25, 2004.
See also the work by Normal Margolus and Tommaso Toffoli.

Added March 31, 2005
See Opinion 57 (November 9, 2003) by Doron Zeilberger: "But, I can hear you retort, this is still just a number, while 4n is, an admittedly easy, but nevertheless a genuine `theorem', valid for all (hence infinitely many) positive integers n. Nonsense! There is no such thing as infinitely many integers (or infinitely many of anything). The formula 4n is ONE fact: it says that for a symbolic n, the number of walks of n steps equals the expression 4n..."

Added May 3, 2005
Laszlo E. Szabo. Formal Systems as Physical Objects: A Physicalist Account of Mathematical Truth, International Studies in the Philosophy of Science, Vol. 17 (2003), pp. 117–125 (preprint: PDF)

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formalism, non-formal, anti-formalist, philosophy, formalist, digital, human, foundations, mathematics, brain, formal, model, modeling, modelling, intuition