Comment by goldenkey | Hacker News Reader

Comment by goldenkey | Hacker News Reader

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Here is the levels of absolute truth in our universe in terms of dependency:

Mathematics > Physics > Chemistry > Biology > Physiology > Medicine

Discoveries in mathematics are truths about the universe. They are deeper than particle physics in some respects. Some parts of mathematics might seem abstract but every mathematical system uses the naturals in its axioms or representation. The naturals are directly based on counting, based on the nature of macroscopic objects in our universe. The universe enforces rules, and the facts about naturals, and systems built upon them, are truths that directly point at the nature of information and complexity in our universe.

Why should mathematicians apologize? Hardy was wrong, mathematics can lead to nukes. But its the base level of truth, there is no other scientific discipline that discerns the patterns of the most abstract physicality - objects, and gleans truths, rules for how objects interact.

Solving the Riemann Hypothesis or other conjectures that aren't even known yet might lead to understandings/models that allow for time machines. It's impossible to know.

But why not seek to understand the universes' laws at its most generic level. Its enlightening. Spiritual. Awakening.

Are you talking about the universe as in the physical universe in which we live ? Because although maths can be used to find out about our physical universe, its abstractions go beyond what is in our physical universe...

Thinking of the maths involved in certain man made games (eg chess), those maths aren't necessarily truths about the physical universe.

Maths have been useful for clear thinking to help understand and predict behavior in the physical world but they remain distinct from the physical.

Isn't chess a part of the physical universe?

I think I understand the point you are making, but it's not such a clear distinction.

Yes it's true chess is part of the physical universe and I can see why it's a difficult point to articulate. Perhaps one way to say it is that the rules of chess and strategy should not depend on the parameters that define your universe, only should depend on the rules of chess you decide to play.

Well, maybe, but it's physically impossible (or at least improbable) for all possible chess games to occur within the physical universe.

maybe if you constrain the game to take place on a wooden board, pieces moved by players. Otherwise, e.g. the ions inside some exotic star may be moving in chess like patterns (for a while). who's to say. Where does the disagreement with the Pythagorean ideal stem from, anyway?

Wait what, I thought it occurred already in some IBM computer, no?

There are estimated to be over 10^120 possible games, so probably not.

You seem to be confusing particle configurations for conceptual truth. Truths about triangles are truths about physical triangles to an extent. Same with circles although you are probably thinking "but theres no perfect circle in Nature!!." True.

But the universe isnt just particle configurations. It's the rules of physics too.

Mathematical truths, the ones mathematicians care about, are founded on axioms that have truth values set by our relationship with the physical world, it informs our intuition. This is inescapable.

I would not disagree that many axioms that we use have direct relationships with the physical world, this is one of many ways math is useful in describing nature. I was trying to say that axioms in general do not have to be set to accommodate the physical world.

Those would be strangely wrong axioms though and probably not what OP had in mind. Just because a theory with axiom 1==0 contains numbers doesn't mean it's math, at least if it is not consistent.

But maybe you are right, math isn't the epitome of human knowledge. It is the art of learning that knowledge. Still, in some way, individual knowledge constitutes the extent of anyone's own universe.

Mathematical truths have nothing to do with the universe.

Extremely naive thing to say. Mathematical truths are merely derived truths about complex structures. These structures borrow intuitive concepts like "sets" or "natural numbers" in their foundations. Mathematics has everything to do with the format of this universe. A statement is only meaningfully true if it is based on axioms that are intuitively true. The direct relationship between what mathematicians agree on as meaningful systems to derive truth from, ie. ZFC , has everything to do with the relstionship between the axioms and intuition that can only be provided by living in this world as a human being.

The old and tired tripe of mathematics being some ethereal detached mental masturbation is so sorely mistaken that I must ask you to respond in as much depth as I did. Because your viewpoint is not only wrong but also the reason so many children grow up to be adults that dislike mathematics. Its a sad disinformation to disconnect the properties of our universe from the science that studies the most abstract type cast of our universe: object.

Natural numbers need not be in scope of intuition. Imagine a diffuse intelligence formed in a fluid (such a thing is at least imaginable), it has no fingers, it recognises no individual things, not even itself, it has no intuition of the natural numbers. If this being makes statement on arithmetic, like 1+1=2, according to you this statement is not "meaningfully true". Yet it is true, and this being has derived it from (abstract) thought. So what is "meaningfully true"?

> need not be in scope of intuition.

> Imagine

Imagination is in principle intuitive.

> it recognises no individual things, not even itself,

self-awareness is part of the common definition of intelligence.

> Imagine [something that is patently made up]

> Yet it is true

utter nonsense

> it has no intuition of the natural numbers. If this being makes statement on arithmetic, like 1+1=2, ...

How?

You can't just say

> imagine Natural numbers need not be in scope of intuition

> then Natural numbers need not be in scope of intuition

I mean, sure you can, but that is circular reasoning and not at all convincing.

You prove his point with your fluid example. Integers have _everything_ to do with our world. If we lived in a fluid world, then the mathematics of that world will not have integers.

Take a look at this: https://en.wikipedia.org/wiki/Mathematical_universe_hypothes...

Perhaps my point is misunderstood here: it is that if we think of mathematical truth as being based on intuition, then since what can be intuited depends on the species, on the society, even on the individual (99% of what was intuitive to Grothendieck is completely beyond me), then mathematical truth becomes relative. I say no, 1+1=2 universally, whether it is intuitive to you or not.

Language too has nothing to do with the Universe - that doesn't prevent sentences in it from "mean"-ing something.

> that doesn't prevent sentences in it

The fact that it could mean the universe or the language pretty much lets your argument fall apart.

Mathmen are more Van Eyk than Pollok.

There is great beauty in this approach to mathematics. It is why I am currently studying mathematics. It is a great defense of why anyone would want to do mathematics.

However, there is another question. Why do we want anyone to do mathematics? Asking, not what math gives the mathematician but instead what math gives society at large. (That is, beyond contented mathematicians).

As long as mathematicians are self-reliant, the second question doesn't matter. However, when mathematicians start making demands of society the second questions is natural.

Note that these questions are orthogonal. Any answer to the mathematics extrinsic value need not inform its intrinsic value. Just like the intrinsic value of mathematics is not related to its extrinsic value.

The point here is that math for math's sake has an awesome byproduct for society: phd's that go commercial. To be afraid that maths will then get squeezed for such phd's to the detriment of it's intrinsic value massively discounts the will of mathematicians and the long view of society. Sure, there will be some that try this squeeze. But their effect will be limited. Meanwhile, this is how mathematics can sustain itself at much higher levels.

Do you really think there is so much true extrinsic value in all the proofs of mathematics? I'd say the value of all those who graduated and went on to do other things easily outweighs all those proofs.

It was the creation and later understanding of those proofs that had extrinsic value. This means that any field of mathematics has value no matter how likely it is to ever be applied.

As such, extrinsically it matters little what field you study, all of it is useful.

"Why should mathematicians apologize?"

Apology in this context means "defense of".

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