## Mathematics (and science in general): Deterministic, or not?

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- adlaiff6
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### Mathematics (and science in general): Deterministic, or not?

I had a conversation today about whether the history of mathematics was deterministic or non-deterministic. Although we never came up with a formal definition of the question, I think it was basically a question of whether there could exist an alien race that followed an entirely different line of discovery and learned entirely different things about the universe, specifically mathematics (because we believe that above all to be universal). Specifically, are the truths we believe to be true (for the most part) universally true, or could there be a completely separate (maybe disjoint) set of truths that are also true? For example, might those aliens have a different Pythagorean Theorem? Might they not deal with numbers at all? Perhaps they base everything on something completely different.

Basically, what do you think?

Basically, what do you think?

- OmenPigeon
- Peddler of Gossamer Lies
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It doesn't specifically address all your questions, but you might want to start with this thread.

As long as I am alive and well I will continue to feel strongly about prose style, to love the surface of the earth, and to take pleasure in scraps of useless information.

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~ George Orwell

- jestingrabbit
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The history of mathematics on this planet indicates that its quite possible imo. Most of the great mathematicians, Archimedes, Euler, Gauss, Newton, Napier etc, were not bound by axiomatic considerations, whereas in the last century that's become a foundation for the subject. It doesn't really effect most of the research that gets done though, so it might not have effected the results that were arrived at.

Even if axiomatic foundations are accepted as necessary, then there are several that are considered suspect, like the axiom of choice and, in some very small groups, the axiom of infinity. There are also things like the continuum hypothesis, which could be accepted or rejected by an alien mathematics, whereas most mathematicians here think its pretty much irrelevant (its independent of the axioms we generally use). Such a decision could have been arrived at if they used the hyperreals as to get a unique set of hyperreals you need to assume it (otherwise different ultrafilters can give you different collections of hyperreal's that don't have an order preserving field isomorphism).

And there could be stuff we have no idea about. It could be a really alien mathematics.

But, in the end, I suspect that we could learn about the other culture's approach, and see the truths that they were describing. It couldn't be so alien that we couldn't come to some mutual understanding imo.

Even if axiomatic foundations are accepted as necessary, then there are several that are considered suspect, like the axiom of choice and, in some very small groups, the axiom of infinity. There are also things like the continuum hypothesis, which could be accepted or rejected by an alien mathematics, whereas most mathematicians here think its pretty much irrelevant (its independent of the axioms we generally use). Such a decision could have been arrived at if they used the hyperreals as to get a unique set of hyperreals you need to assume it (otherwise different ultrafilters can give you different collections of hyperreal's that don't have an order preserving field isomorphism).

And there could be stuff we have no idea about. It could be a really alien mathematics.

But, in the end, I suspect that we could learn about the other culture's approach, and see the truths that they were describing. It couldn't be so alien that we couldn't come to some mutual understanding imo.

This page may be of interest: http://www.maths.nott.ac.uk/personal/anw/Research/Hack/

Patashu wrote:Mathematics could have been completely different.

But "science in general" couldn't; that's a model of the universe, so while it could be differently described, it would have to be a pretty similar model if it is to describe the same universe.

It's the same with Pythagoras: Aliens could never prove that a^2+b^2=c^3, because it doesn't equal that, at least, not in Euclidean geometry, but they could perhaps come up with another way to calculate c that produced the same results.

But I reckon (through gut feeling, mostly) the smart money would be on them having much the same maths as us, at least at a basic level. Of course, they could be huge and fast or tiny and fast, and live in a more relativistic or quantum world. Then they'd be really interesting.

They would have had just as much time as us (possibly more or less) to evolve their mathematics.

@ everyone else

It's hard to say whether they could have a different set from us. As said previously a^2+b^2 != c^3 , but that doesn't mean everything is set in stone. A lot of physics is based on assumption and reference. The aliens would be living in a reference frame much different from our own. This would certainly make their discoveries much different. Some discoveries would be favored, whilst others not. Gravity may or may not be discovered. This would most likely lead them in opposite directions as to what has been discovered in our world.

These are just extreme basics and I'm over simplifying, but what I'm trying to say is that I have no idea.

I think their mathematics would be equivelent to our own. Maybe definitions would be different (think languages).

- gmalivuk
- GNU Terry Pratchett
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I imagine there'd have to be enough correspondence between the mathematics of any sufficiently advanced (i.e. spacefaring) species for communication (at least of mathematical principles) to be possible.

Seems certain numbers like e pop up so often that it'd be hard not to stumble across it at some point, for example.

Seems certain numbers like e pop up so often that it'd be hard not to stumble across it at some point, for example.

- adlaiff6
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@EstLladon: For future reference, I, like others, am disinclined to listen to you when you shout at me. Of course, if you happen to be typing on a UNIVAC, I would understand, and immediately ask how you got the rest of the characters in lower-case.

Less frivolously, I'd like to offer my thoughts now. First, I believe truth to be deterministic, but the sequential study of mathematics to be non-deterministic. I'm pretty sure we're slowly converging on some finite set of true statements, but there are many, many ways to get there. For example, both Pythagoras, in Greece, and someone else (forget who), in India, came up with the idea that a^2 + b^2 = c^2, but Pythagoras did it by looking at geometry, while the Indians did it with algebra (I think). Thus one might say that the Greeks at some point probably had geometry far advanced in comparison to the Indians, and vice versa for algebra. However, through each line of study, they both realized one and the same true statement.

Alternatively, I think there also just might be some "higher order" truth (no, not religion, sorry kids). Think for a moment about the Halting Problem (or wikipedia it if you can't yet). Suppose there were a machine to determine if a Turing machine halted on some input. Then, that machine would be able to compute things about Turing machines, and would therefore be of "higher order" than a Turing machine (call it a super-Turing machine maybe?), and along with it would come a whole class of things that would be computable on a higher level. Now there could be this higher order of computability, and there might be truths of this order entirely separate and perhaps contradictory (not sure yet on that one, but we might see) to those we normally deal with. Of course, the problem with this idea is that we can then generalize to an infinite hierarchy of "(super-)^n Turing machines", each of which having a higher computational power than the last. Here we run into a very difficult thing to think about, and most would probably say it means God exists or something equivalently silly.

Sorry about my bias against standard religion. It is not of malice.

Less frivolously, I'd like to offer my thoughts now. First, I believe truth to be deterministic, but the sequential study of mathematics to be non-deterministic. I'm pretty sure we're slowly converging on some finite set of true statements, but there are many, many ways to get there. For example, both Pythagoras, in Greece, and someone else (forget who), in India, came up with the idea that a^2 + b^2 = c^2, but Pythagoras did it by looking at geometry, while the Indians did it with algebra (I think). Thus one might say that the Greeks at some point probably had geometry far advanced in comparison to the Indians, and vice versa for algebra. However, through each line of study, they both realized one and the same true statement.

Alternatively, I think there also just might be some "higher order" truth (no, not religion, sorry kids). Think for a moment about the Halting Problem (or wikipedia it if you can't yet). Suppose there were a machine to determine if a Turing machine halted on some input. Then, that machine would be able to compute things about Turing machines, and would therefore be of "higher order" than a Turing machine (call it a super-Turing machine maybe?), and along with it would come a whole class of things that would be computable on a higher level. Now there could be this higher order of computability, and there might be truths of this order entirely separate and perhaps contradictory (not sure yet on that one, but we might see) to those we normally deal with. Of course, the problem with this idea is that we can then generalize to an infinite hierarchy of "(super-)^n Turing machines", each of which having a higher computational power than the last. Here we run into a very difficult thing to think about, and most would probably say it means God exists or something equivalently silly.

Sorry about my bias against standard religion. It is not of malice.

- gmalivuk
- GNU Terry Pratchett
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adlaiff6 wrote:Of course, the problem with this idea is that we can then generalize to an infinite hierarchy of "(super-)^n Turing machines", each of which having a higher computational power than the last. Here we run into a very difficult thing to think about, and most would probably say it means God exists or something equivalently silly.

Such hierarchies already exist for formal logical and mathematical systems. The general trend is that, the more powerful our system is (i.e. the more things it can talk about), the less "nice" it is. First-order logic is not complete. That's fine, neither is arithmetic. Second-order logic, in which you can quantify over predicates (analogous to mathematical considerations of sets of relations and functions) is neither sound nor complete. I don't know anything about third-order logic (in which you can quantify over the properties of first-order predicates), but it's probably even more pathological.

So it's sort of like, the more you can do with a system, the less sure you can be about its results. So I guess if this means something about God, it means God may exist but we can't know anything about it. Which is for all practical purposes as if God doesn't exist. (Pascal's Wager argument, if really run from the premise of complete agnosticism he claims, really ends up seeming to say you should act as though there is no God and all the costs and benefits of a decision are those that occur here in this life.)

- adlaiff6
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Heh, just poking fun. Don't be sorry.EstLladon wrote:@adlaiff6: I'm sorry, I didn't mean to shout, I just didn't came up with another text-based way to highlight some statements. I'm sorry.

By the way, what is truth?

Dunno. If you find out, call me up.

gmalivuk wrote:Such hierarchies already exist for formal logical and mathematical systems. The general trend is that, the more powerful our system is (i.e. the more things it can talk about), the less "nice" it is. First-order logic is not complete. That's fine, neither is arithmetic. Second-order logic, in which you can quantify over predicates (analogous to mathematical considerations of sets of relations and functions) is neither sound nor complete. I don't know anything about third-order logic (in which you can quantify over the properties of first-order predicates), but it's probably even more pathological.

So it's sort of like, the more you can do with a system, the less sure you can be about its results. So I guess if this means something about God, it means God may exist but we can't know anything about it. Which is for all practical purposes as if God doesn't exist. (Pascal's Wager argument, if really run from the premise of complete agnosticism he claims, really ends up seeming to say you should act as though there is no God and all the costs and benefits of a decision are those that occur here in this life.)

Sounds like some kind of twisted Heisenberg or something.

You're wrong about Pascal though, he said you should believe in God, or at least pretend to, under the assumption that the expected return was infinite in that case, and bounded otherwise.

- gmalivuk
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adlaiff6 wrote:You're wrong about Pascal though, he said you should believe in God, or at least pretend to, under the assumption that the expected return was infinite in that case, and bounded otherwise.

Right, but that assumption is baseless, since he previously claims we can't know anything of the nature or the extent of God. Then goes onto argue that if God exists, it must be the Catholic version.

But under his earlier presumed agnosticism, we can't know that, if God existed, the payoff for believing would be infinite. Why not a God who prefers reason, and thus awards those who don't believe things with no evidence? Or a malevolent God who rewards nonbelievers and sends the faithful to hell? Or an even worse Got who just sends everyone to hell because he's a lazy fuck and would prefer to let Satan deal with the dead?

If you truly can't know anything about God, you can't rationally base any decisions on the level of utility they might provide after death. Because you can't know (or even guess) anything about what happens after death. In which case you should base your decisions on what you can guess, which is what happens here during this life, which can be explained without any reference to God at all. So in other words, running that kind of argument from the first assumption he makes instead of the second, you ought to believe (or at least pretend to) that there is no God.

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