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/sci/ - Science & Math


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>> No.5950573 [View]
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I'm extremely uncomfortable with this classical presentation of classical mechanics. In general, I think this "3 axioms" way of introducing mechanics leads to much confusion, as it gives a way to think of what a physical theory is, which then doesn't match up with how general relativity is presented. Specifically, "Newtons first axiom" suggest an interpretation for what a physical axiom is supposed to do and provide which is more unmathematical and connected to the world than any of the definitions and axioms of mothern theories such as quantum mechanics or the Lagrangian/Hamiltonian formulation of classical mechanics and general relativity.

>> No.5840569 [View]
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What are some nice adjoints in differential geometry?
I don't quite have a feel for the former, and I'm pretty sure a geometrical example would help.
Also, more abstractly, how do adjoints relate to (the syntax of) predicate logic. I hear they make structural truths about quantifiers more clear.

>Still living under a set-theoretical rock
Why can't we all be friends?
On a serious note, I completely agree that making functions (and thereby associativity if you want) a primary part of your axiomatics.
At the same time I really seek to understand understand why you would want to drop such a fruitful notion as the binary predicate of membership. In short, I don't see why people would want to axiomize topoi in a standalone category theoretical framework, if there seems to be no harm in having at least a weak set theory on the side.
Is it only philosophy?

>> No.5712418 [View]
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Form a 1870 perspective, the applicability of kinetic gas theory and the explainations for behaviour of chemicals.
And once you have particle accelerators, it's hard to deny that you have some sort of idea of the things your work with.

>> No.5697912 [DELETED]  [View]
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>In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that "the product of a collection of non-empty sets is non-empty"

I consider this formulation polemic.
You have a bunch of other definitions in the background together with a large universe, this is why this happens.
The construction of the product is a set (per definition, so it IS a set in any case) whose elements are such and such. In your set theory without choice, "the product of non-empty sets results to be an empty set" is true, but only because the defintion requires you to collect the elements the set is supposed to contain, and then, without the axiom of choice, you can't find any. Hence the definition returns the empty set.

To say
"In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that 'the product of a collection of non-empty sets is non-empty' "
subliminally suggests that if you negate the axiom, then the theory is more well behaved - yes, but only because you robbed it it's power.

As far as opionions go, I' currently in a melancholy mood, telling me to not care too much about the man made desire to have a single framework for foundation.
That is I learn towards computation and complexity considerations now.

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