Karlis Podnieks

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

Digital Mathematics and Non-digital Mathematics

Trying to understand anti-formalists

By K. Podnieks

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

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

Mathematicians are like mad tailors: they are making "all possible clothes" and hope to make also something suitable for dressing...

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

Gordon Fischer: 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 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. Each physical phenomenon is, of course, a very precise "model" of itself. Do we need more than this? I.e. do we need non-digitalizable models at all?
4. For me as an old Hilbert style formalist, the formalizable (i.e. digitalizable) mathematics is the "only true" kind of mathematics. But, may be, the "occultist style" opponents of 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?

Any comments are welcome - click here.

October 4, 1999

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

Writings:

Hyper-textbooks for students in mathematical logic
[University of Latvia, Institute of Mathematics and Computer Science]:

1. Introduction to Mathematical Logic, by Karlis Podnieks, Dr. Math. and Vilnis Detlovs, Dr. Math.
2. Around Goedel's Theorem, by Karlis Podnieks, Dr.Math.
   My Main Theses: I define mathematical theories as stable self-contained systems of reasoning, and formal theories - as mathematical models of such systems. Working with stable self-contained models mathematicians have learned to draw maximum of conclusions from a minimum of premises. This is why mathematical modeling is so efficient.
   For me, Goedel's results are the crucial evidence that stable self-contained systems of reasoning cannot be perfect (just because they are stable and self-contained). Such systems are inevitably either very restricted in power (i.e. they cannot express the notion of natural numbers with induction principle), or they are powerful enough, yet then they lead inevitably either to contradictions, or to undecidable propositions.

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