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

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**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
http://doug-pc.itp.ucsb.edu/online/plecture/penrose/
(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 http://web.mit.edu/philos/www/jjt.html

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**REAL** REAL WORLDS (Physical and
MATHEMATICAL) ARE DISCRETE.

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 4^{n} 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 4^{n} is ONE fact: it says that for a *symbolic*
n, the number of walks of n steps equals the *expression*
4^{n}..."

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