Hacker News Reader

Comment by barrkel | Hacker News Reader

I think it's a system of symbols and rules. By mechanically provable, I mean that given axioms (assumptions) and rules, you can devise a machine (i.e. something which follows rules, with no independent thinking or homunculus) which generates statements which follow from the axioms and rules, and this is what "true" means in the system.

View on HN

So would you say that your system of symbols and rules is real? Could it be that we both use the same system of symbols and rules, with the same assumptions, but derive different conclusions? If not, why not?

if the system is sound, then (I think) by definition you cannot prove different (wrong) conclusions.

if you derive a different result, by soundness those would be equivalent ???

It does not really matter if the system is sound or not, right? Although of course a sound one is far more interesting. Anyway, any way of justifying this is mathematical (and so would be the definition of soundness, if it was relevant here). If math is not real, then there is no justification.

math describes (fragments) of reality;

therefore it is of no consequence if math as itself is "real" or not. it is intended to model whatever "real" even is.

somewhat similarly: in modern logical theories whatever "true" (and/including "false") even mean doesn't matter. is left out of the logical theory and it is effectively a mere parameter.

all the subject does is gurantee "truth in, truth out" (and complementarily "false in, false out")

the precise details of true "and/including" false, seems to me, are somewhere in the boundary between "classical" and "intuitionism" (or "constructivism")

the subtle distinction between intuitionism" and "constructivism" is above my pay grade (and seemingly above the paygrade of everybody I've had the chance of discussing this with)

> math describes (fragments) of reality;

This is only possible if math itself is real. Note that I am not saying that a particular axiom system like Euclidean geometry has some sort of "real physical manifestation". No, what I am saying is that logical reasoning itself is real. And our reasoning about logical reasoning is certainly real as well, even if logical reasoning itself happens in very abstract form. Math itself might be viewed by some as just a game of symbols. But that doesn't change the fact that the game itself is real. Would it be otherwise, then math would be about as important as chess.

I like to draw a distinction between real and ideal.

I insist that math is ideal. it models reality ideally.

this distinction is important because otherwise we mix together something, and the ideas and concepts (e.g. symbols and rules) we use to describe and model said something.

the game is not real. people playing the game are real, the game getting played is real. the game on its own as may be described in symbols is ideal.

i suppose what it all is all about is the intersection between this reality and this ideality.

You can say that a certain axiom system models a certain part of reality in an ideal way. But whatever is ideal, is also real, because otherwise there is nothing that could model anything. So your intersection of reality and ideality is just ideality itself.

It has no substance other than its representations; there's nothing of it you can touch which is physical. At best, there is a correspondence between the physical and the system.

I agree with you here, at least in the sense that the system is definitely not physical. You didn't answer my question, though. Is it real?

All information that exists is physical, encoded in a physical substrate. Beads in an abacus, holes in a punched card, distributions of charge in a computer memory. Hypothetical information that is not encoded physically cannot be causal. A book or computer program that have not been written can have no effects. Only information that exists in a physical encoding can be causal, by virtue of it’s physicality.

Not sure where you are going with this. What is causality? Is there a non-mathematical way of making it precise? And if I have found some way to make it precise, does it matter if I write it down here in this HN comment, or on a piece of paper, or just think it? Is the mathematical content different depending on how it is expressed in physical reality?

Wow, you excavated an ocean to cover a puddle. how this: "something which follows rules, with no independent thinking or homunculus" can be easier to prove and reason than the initial rules?

For example, what is simpler to reason out: does a chess move violate chess rules, or, there exist a method to construct an electomechanical device, that will Correctly determine whether this chess move is legal?

The reason I brought up a machine explain something as being mechanical is to clarify that it doesn't require intuition.

We can make machines that count. A trivial example: pebbles in a bucket. Neither the pebbles nor the bucket need intelligence to act as (have a correspondence with) a counter.

Completely agree about the machines. Many mathematical results become more rigorous as a result of "can be done on this kind of machine" type of proofs. However, if the goal is getting rid of intuition, machines don't help! Because no matter what the machine (bucket of pebbles or a Buchholz hydra), it takes a lot of intuition to prove, that particular machine correctly enforces intended rules. Usually more intuition, than the original problem.

Without having proof for the machine itself that machine is declared axiomatic. It is a valid way to go about things of course, but I would hesitate calling it "not requiring intuition".

Reply on news.ycombinator.com