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Comment by logicallee | Hacker News Reader

When you're a mathematician, everything is black or white, but you get to define what those two words mean.

I think that's another misunderstood thing about math.

By the time you get into higher mathematics, you get to a place where nearly everything is unknown. I remember reaching the level of ordinary differential equations and realizing that "most" non-linear equations had no exact solution at all.

In ways, math and programming have this in common - as the size of the systems being studied increases, a randomly chosen example becomes harder and harder to deal and thus the "art" consists in choosing important and tractable examples out of a world of huge but intractable systems.

When I was 17, high school, and really enjoying maths a maths professor came to our school to speak to a few of us about continuing maths at University. She really annoyed me. She asked what about maths we liked and a friend and I answered that it was the certainty - that things were right or wrong and you could most often tell plainly which they were.

She of course responded that we couldn't be more wrong.

Annoyed the hell out of me until I came across Godel and began to understand maths from a different perspective.

Wish I'd had chance to study under that professor.

> I remember reaching the level of ordinary differential equations and realizing that "most" non-linear equations had no exact solution at all.

This may be a bit theoretical for most readers, but every ODE that has one and only one solution has an exact solution. That the best definition for that function is "the solution to that ODE" does not mean that the function is any less exact than something you can construct out of exp, sin, cos, etc. In fact it would be completely reasonable to define exp as a solution to a differential equation and then to prove afterwards, that exp is equivalent to the usual power series and accidentally also has some other nice properties :).

Ah,

I spoke too loosely. I should have written "closed-form symbolic solution" rather than "exact solution" (though "exact" is used colloquially to mean this, it's not clear). Yes, all solutions to an ode would be "exact solutions" in the literal sense.

It is worth noting also that an ode having a solution around a point doesn't imply that it can be extended indefinitely - you can hit "singularities" that make extension impossible.

every ODE that has one and only one solution has an exact solution

Upvoted, but - is that a particular theorem, or are you just basically stating the definition of "having a solution"? Wouldn't your statement be true of any system of differential equations, not just ODEs?

I just stated the definition of having one and only one solution, yes :). That was kind of the point I was making.

I originally intended to write a statement specific to ODE, but was then to lazy to look the specifics up ^^.

I read logicalee's comment as a reference to the way most (all?) branches of mathematics may be built using sets and their axioms. Set theory boils down to "in the set" and "not in the set", ie. black and white.

Quite interestingly, the axioms mathematics is build upon do not let you decide every question that you might come up with. As such in mathematics not everything is black or white. For example, the continuum hypothesis is independent of the most commonly used axiom system ZFC. To explain what that means imagine ZFC would be describing an apple. Out of the definition of "apple" you can get theorems that tell you "if you start on any point of the apple and dig down in a straight line, at some future point in time you will reach the 'other side' of the apple" or some statements you can disprove "if you walk on the surface of the apple in a straight you will never come back to point where you started". What you will not get is the colour of the apple. Everything you know about the apple is consistent with it being green. But all that is also consistent with it being red. The colour of the apple is independent of everything that you are interested in in an apple. Therefore there is no reasonable answer to the question "which colour does the apple have", except maybe: It should not matter. The same is true of the continuum hypothesis. I remember reading that if you need the CH to prove something, then you should definitely reconsider the statement you are trying to prove :).

> I remember reading that if you need the CH to prove something, then you should definitely reconsider the statement you are trying to prove

This is not entirely true. There are many proofs that begin by assuming CH. There are also two ways of interpreting "need CH to prove". One way is that the thing you are trying to prove is equivalent to CH, which is an interesting/useful result. Another way to interpret it is that you can not come up with another way of proving it. In the latter case, the CH based proof justifies that your proposition is not inconsistent, and may even lead you (or others) to proof that it is true regardless of CH. If I recall correctly some statements have been proved by proving the statement when CH is true, and also when CH is false.

Ah but the real story is that like sophisticated math systems, in relation to any fixed axiomization, set theory involves provably, provably false and unprovable/independent statements. And in relation to any fixed model, set theory involves true and provable, true but unprovable, false but un-disprovable and false and disprovable.

And when you're math as a human endeavor, you can also add "provable but not yet proven" and "proven independent"

So, learn some stuff, see how far from black and white higher math can be.

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