originally posted in:Secular Sevens
It's my understanding that mathematical truths [i]do[/i] in fact rely on the physical laws of the universe, in the sense that science has not excluded the possibility of there being other universes with different physical constants and different mathematical truths from our own. Things in math can only be said to be absolutely true in our universe because of the fundamental underpinnings that comprise it.
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But isn't a mathematical truth such as ''5 + 7 = 12'' based solely on the definitions of 5, 7, 12, addition, and equality, as opposed to being (partially) based on conditions of the physical world? Sure, there could be worlds so different from ours that our mathematical notions would fail to meaningfully describe it, but whether a mathematical truth is useful has no bearing on whether or not it's true.
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Edited by Ric_Adbur: 8/7/2013 6:04:44 AMThe concept of "five" makes sense to us in our reference frame (4-dimensional space-time), but who is to say that it would continue to do so in another? Do we really have the knowledge to claim with certainty that mathematics as we understand it is an unwavering constant in all possible realities, in everything that exists, when we are only at the very beginning of our research into the multiverse concept? I mean, we often perceive things as being separate that aren't necessarily so - quantum superposition, for example; something can physically exist simultaneously in many seemingly-opposing states. We would think of different states as being mutually-exclusive from our reference frame, but when you go small enough that ceases to be the case. Perhaps if you look from far enough back, other things we think of as true cease to make sense? If a world is so different that our mathematical notions cease to describe it, isn't that the very definition of those notions being untrue for that world? Hell, we don't even fully understand our own world - perhaps, like with quantum superposition, at different reference frames in our reality, mathematical constants as we know them cease to function in the same way. If a person could exist at the quantum level, they would surely find it difficult to wrap their head around the functionality of our reference frame, perhaps the same holds true for other reference frames we cannot yet conceive of. Perhaps we are even fundamentally incapable of conceiving of them - just because we haven't yet come up against a problem that proved too much for our intellect doesn't mean we won't at some point. We may simply lack the intelligence to fully comprehend the world around us.
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[quote]The concept of "five" makes sense to us in our reference frame (4-dimensional space-time), but who is to say that it would continue to do so in another?[/quote]What do you mean ''makes sense''? AFAIK you could either mean that its definition is logically coherent, or that it meaningfully represents reality in some manner. If you're saying the latter: whether or not it is useful or meaningful is only relevant to whether or not we should use the concept. The definition itself is self-sufficient. That is, unless, you mean the former; then I don't see why the logical consistency of the concept of 5 (or any concept, really) is based on contingent properties of reality or our sensory experience. [quote]Do we really have the knowledge to claim with certainty that mathematics as we understand it is an unwavering constant in all possible realities, in everything that exists, when we are only at the very beginning of our research into the multiverse concept?[/quote]Well if the truth value of a statement like 5 + 7 = 12 is based [u]only[/u] on the definitions of the involved concepts, then it's basically a truism to say that it's not based on contingent factors of reality. To illustrate my argument more clearly: P1: The truth of '5 + 7 = 12' is based only on the definitions of 5, 7, 12, addition, and equality P2: The definitions of these concepts are valid regardless of any contingent factors of reality (e.g. nothing could just change, and then make our definition of 12 invalid) P3: The definitions of these concepts are not contingent factors of reality (follows from P2) C: The truth of '5 + 7 = 12' is not based on contingent factors of reality As I understand, the premise you take issue with is P2. Please note that when I say that a concept is 'valid,' I mean that it does not entail any logical contradictions. Now, I could imagine a world in which these concepts (5, addition, etc.) were irrelevant and useless, but not one where they entail contradictions. While certain behaviors and properties of reality give rise to the use of certain concepts, this doesn't imply that the concept's validity is contingent upon these behaviors. For example, we only have a concept of a 'frog' because we observe them. However, if we were in a world with no frogs, the concept would be no less valid, even if there were nothing to cause us to conceive of frogs. The same should go for the more abstract concept of units; even in some strange world where there are no 'units' of anything, this would not make the concept invalid. [quote]I mean, we often perceive things as being separate that aren't necessarily so - quantum superposition, for example; something can physically exist simultaneously in many seemingly-opposing states. We would think of different states as being mutually-exclusive from our reference frame, but when you go small enough that ceases to be the case. Perhaps if you look from far enough back, other things we think of as true cease to make sense?[/quote]The truths you mentioned only cease to make sense if we (erroneously) extrapolate our experience of the macro world to the quantum world. The fact that two different positions for an object are mutually exclusive of the macro world is no less true from the quantum frame of reference, so long as we keep in mind the ''of the macro world'' clause. Put more clearly, all you've shown me is that the macro world and the quantum world have very different behaviors, not that certain fundamental truths of the macro world cease to make sense from the frame of reference of the quantum world. Sure, quantum behavior is counterintuitive from our frame of reference, but intuition is only loosely associated with truth. [quote]If a world is so different that our mathematical notions cease to describe it, isn't that the very definition of those notions being untrue for that world?[/quote]Just because a model isn't useful doesn't mean it is logically invalid. Take, for example, [url=http://en.wikipedia.org/wiki/Euclidean_geometry]Euclidean geometry[/url]. Basically, it described geometry on a flat plane, and it formulates theorems based on five axioms/postulates. So, a theorem would take the form of ''If the five postulates hold, then the Pythagorean theorem is true.'' Unfortunately, according to Einstein's theory of general relativity, our universe is, in some parts, distorted and curved. If you were to draw a right triangle in the real world with one vertex extremely close to a black hole, the Pythagorean theorem would not hold. Nevertheless, the statement ''If the five postulates hold, then the Pythagorean theorem is true'' is still true. It's not useful (and it doesn't describe reality) if you're doing geometry close to a black hole, but it's still a truism. And essentially, all mathematical truths are just like this, taking the form of ''If these axioms are true, then this statement is true.'' (the axioms we use are [url=http://en.wikipedia.org/wiki/ZFC#The_axioms]here[/url], in case I've sparked an interest in formal math for you :D) Now if you'd consider a mathematical truth to only be 'true' if it is also useful and describes reality in some manner, I'd concede that you're correct. But I'd also consider that a rather limited notion of ''mathematical truth,'' since basically all of modern mathematics is based on previously described axiomatic systems (that is to say, a theorem is a 'mathematical truth' if the proof for it is valid, regardless of whether or not it is useful).
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Edited by Eternal Way: 8/7/2013 7:01:07 PMScrew the Aristotelian criteria for definitions. :D
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Hmmm... maybe 2:30am isn't the best time to try to wrap my brain around this stuff. lol
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Pansy, you'd never survive a 48-hour philosophy marathon.