Wednesday, March 20, 2024

Can you feel the machine?

In the recent academy award winning movie Oppenheimer, Niels Bohr tests a young Oppenheimer.
Bohr: Algebra's like sheet music, the important thing isn't can you read music, it's can you hear it. Can you hear the music, Robert?
Oppenheimer: Yes, I can.

I can't hear the algebra but I feel the machine.

A Turing machine has a formal mathematical definition but that's not how I think of it. When I write code, or prove a theorem involving computation, I feel the machine processing step by step. I don't really visualize or hear the machine, but it sings to me, I feel it humming along, updating variables, looping, branching, searching until it arrives at its final destination and gives us an answer. I know computers don't physically work exactly this way, but that doesn't stop my metaphorical machine from its manipulations.

I felt this way since I first started programming in Basic in the 1970's, or maybe even before that on calculators or using the phone or sending a letter. I could feel the process moving along.

So when I took my first CS theory class, I could feel those finite automata moving along states and the PDAs pushing and popping. I even felt the languages pumping. After graduate complexity, I knew I found my calling.

And that's why I prefer Kolmogorov Complexity arguments more than information theory. Information doesn't sing to me but I can see the compression.

As computational complexity evolved, focusing more on combinatorics, algebra and analysis, and often on models that have little to do with computation, I sometimes find myself lost amongst the static math. But give me a Turing machine and I can compute anything!

1. "... Information doesn't sing do me but I can see the compression."
sing to me or do me?
Is this a typo or I'm missing something? located at (n-1)th paragraph.

1. I meant "to me". Fixed.

2. 'Feeling the machine' is awesome. I believe this 'operational' way of doing mathematics on a completed infinitely large object/tape/matrix (that is, a procedural way of thinking, yet with a static ontology) is due to Turing et al., and especially Cantor.

In my understanding, Cantor's famous diagonal argument contains a procedural component that explains how to construct the anti-diagonal. It pertains to a completed matrix, which represents a full-fledged bijection between finite strings and bit sequences. This, essentially, presents a Platonic backdrop.

So, arguably, we see Cantor -- and, later, Turing even more so -- apply causal reasoning (= feeling the machine) in a non-causal (Platonic) landscape.

Fusing the causality of a (mathematical) machine with the non-causality of (mathematical) Platonism is something Cantor was permitted to do, since he had idealistic leanings -- according to at least one Cantor scholar (i.e., Jane).

We need the static backdrop to diagonalize out of a completed class of, say, linear bounded automata. We employ the procedural approach to construct a mathematical machine situated beyond the full-fledged class.

This post wonderfully captures the essence of abstract mathematics in comp. science.

1. "Cantor's famous diagonal argument contains a procedural component"

All mathematicians are somehow "cheating", they use procedural arguments while pretending that maths ontology is timeless.

This is especially clear with Russel paradox:
Okaaay, here is the set of sets which don't contain themselves (which sounds a pretty natural concept), Oh! Dang! Where do I do I stash it now?

CHEATING with time!!!

Without such "cheating" you could say goodbye to all diagonalizations, including Godel incompleteness and the halting problem. :-D

2. I don't disagree at all. I eschew the word 'cheating,' as I don't want to choose sides when studying the history of maths.

I believe Brouwer would have agreed with the "cheating" connotation; he went for purely procedural maths only (to put it very crudely). Wittgenstein and Poincare are other examples.

As a comp. scientist, however, I can appreciate the many diagonal arguments in the literature.

3. Diagonalization is a constructive process. Going from double negative to positive is the nonconstructive part.