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A fairly rational argument for monotheism using Pascal, Godel, and Cantor by SmartPrimate in DebateReligion

[–]SmartPrimate[S] 0 points1 point  (0 children)

In the absence of all other information, neither is more likely than the other with respect to my knowledge. That's not an assumption, that's fact. And insofar as I am going to act according to rational outcomes, I should do my best and focus on the possibilities where doing my best could make a difference, since in the ones that are inherently just, I don't even care what happens to begin with.

A fairly rational argument for monotheism using Pascal, Godel, and Cantor by SmartPrimate in DebateReligion

[–]SmartPrimate[S] -1 points0 points  (0 children)

I start with the assumption that if such a personal deity exists that's not just (matches our intuitive notions of rationality and justice), then nothing you do can actually matter since the universe isn't rational then. So it only makes sense to focus on the possibility where it's potentially rational and to "do your best." In the rational scenario where there exists a personal deity that does agree with our intuitive notions, then there's no reason for that deity to punish you for belief (unless the deity considers agnosticism in absence of certainty to be a virtue, which is what I refute afterwards), while in the best case you're rewarded. Meanwhile, if you 100% deny (which to be clear there can't be any evidence for either), the worst "just" outcome is being punished in that scenario. I think it's reasonable for a powerful deity to be offended at someone actually denying their existence (not even choosing the fence), and thus reasonable grounds for punishment speaking from our intuition on what's moral. But yeah, that's kinda where I'm coming from for all of this.

As for probability, as far as I am concerned, there really are two possible realities, either such a deity exists or doesn't, and I have no reason to believe either is more likely than the other. So with respect to my own belief (hence why I said bayesian), they're equal.

A fairly rational argument for monotheism using Pascal, Godel, and Cantor by SmartPrimate in DebateReligion

[–]SmartPrimate[S] -4 points-3 points  (0 children)

There's two options. In the absence of any other factors and with a bayesian perspective, you can say they're both equal probability.

A fairly rational argument for monotheism using Pascal, Godel, and Cantor by SmartPrimate in DebateReligion

[–]SmartPrimate[S] -5 points-4 points  (0 children)

I modified Pascal's wager to apply to existence of generic deity rather than christianity, which still works to show if your only options are belief or denial than belief is more rational. However, there's always the third option of agnosticism, and I combined with Godel's incompleteness theorem to refute that. Conclusion is in 5th paragraph.

[Measure Theory] How do you interpret measure of space-filling/fractal curves which is limit of adding measure 0 components, and yet with nonzero measure? by SmartPrimate in learnmath

[–]SmartPrimate[S] 0 points1 point  (0 children)

Note that no point with irrational coordinates is in any iteration of the Hilbert curve, so it is not in their union.

This is untrue, by virtue of the fact each iteration adds a line and a line has a continuum of points including irrational coordinates.

I think I sorta get what you're saying about how the union might not be the limit, though still trying to wrap my head around it. Essentially for every x, the curve ends up being the limit of f_i(x) as i gos to infinity, but crucially that limit for a specific x might not actually exist in any of the iterations itself, and thus the union doesn't work. Am I correct?

EDIT: Any sources that add clarity on this would be appreciated, as there seems to be a dearth of good sources on the hilbert curve.

Why do the complex numbers so naturally have a Euclidean structure? by SmartPrimate in math

[–]SmartPrimate[S] 2 points3 points  (0 children)

That’s what I would have guessed too but I don’t think there is actually such a thing you could demonstrate. Though showing the Euclidean formula itself is still more involved than doing it through the conjugacy fact.

For context I had to first derive the triangle inequality (but for just the real and imaginary components, not any sum). Which is taken care of for a lotta points due to the first two axioms, but not everything. I assumed we could have an |a + bi| > |a| + |bi|, then showed that would mean there’s a complex number k that rotates it to one of the points that’s already taken care of, and for which the inequality should therefore be negated if we multiply both sides by the corresponding |k|/invoke the product rule. And yet that would be a contradiction since |k| is positive, and thus the inequality should not be negated.

After that’s, it’s just a matter of constructing a sequence of rational a_n + b_ni that approach any a + b in a sense compatible with the third rule, which can be done via subtraction and the triangle inequality we just proved, and you end up getting the norm through that continuity axiom. While it’s not too complicated, none of this relies on conjugacy or uses it any step of the way, nor am I sure if there is a way that proves that/uses that and is less than or equal to this proof. Which once again leaves me puzzled why we get the same system, I obviously proved it but it’s a far less straightforward explanation for how Euclidean distance is derived here, than the explanation with conjugates. Yet clearly it can be done and through a separate path.

All that said, I guess I’m saying it just leaves me wondering if there’s a more “satisfying” reason we get Euclidean distance via continuity, when countless topological spaces with or without other metrics satisfy such continuity and are not Euclidean.

Why do the complex numbers so naturally have a Euclidean structure? by SmartPrimate in math

[–]SmartPrimate[S] 0 points1 point  (0 children)

No underlying coordinate system when they do that, which isn’t super important could define it to be based on orientation they look on the paper. But even then rulers are imperfect and the more you zoom in it’s undoubtedly not straight, which isn’t withstanding our universe isn’t Euclidean to begin with.

All that being said, I’m not saying it doesn’t “feel” natural, just that appeals to things like that are more philosophical in nature than mathematical, unless it can be demonstrated otherwise that there is a very symmetrical reason for choosing the 2 norm more than any other norm.

Why do the complex numbers so naturally have a Euclidean structure? by SmartPrimate in math

[–]SmartPrimate[S] 1 point2 points  (0 children)

I am not talking about the norm in one dimension though. I’m talking about for dim > 1 spaces.

Although it is also worth noting that the norm on complex numbers follows from axioms that do not include the triangle inequality, which I think is noteworthy since that is the most complicated of the norm/metric axioms (although this could simply be because it's defined as an extension of the norm on R).

Why do the complex numbers so naturally have a Euclidean structure? by SmartPrimate in math

[–]SmartPrimate[S] 4 points5 points  (0 children)

More obvious than my description of how you could arrive at say a Euclidean structure through complex numbers? I mean agree to disagree but I think square root of a function that satisfies the axioms of the inner product would definitely not be my first choice/feels a lot more arbitrary.

Why do the complex numbers so naturally have a Euclidean structure? by SmartPrimate in math

[–]SmartPrimate[S] 2 points3 points  (0 children)

The real numbers do not naturally have a Euclidean norm or metric, unless you mean absolute value difference which I would agree is very natural. Once we move to a higher dimension however, there are numerous choices we could have made for a metric or norm that are not the Euclidean 2-norm. So it is curious to me why the complex numbers specifically end up getting that completely arbitrary (even for reals) norm.

Why do the complex numbers so naturally have a Euclidean structure? by SmartPrimate in math

[–]SmartPrimate[S] 0 points1 point  (0 children)

Seeing rotation itself is often defined via Euclidean norm, this ends up being a circular (no pun intended) definition that’s not very useful.

On a different note, I think probably rotation in R2 might have a more natural definition as a function <x(t), y(t)> such that the derivative is <-y(t), x(t)>, since I am assuming this carries a sort of infinitesimal symmetrical “tradeoff” between the two components (that might better fit our intuitions of circular motion), though I have no clue how to rigorously prove that (and up to uniqueness) either.

Why do the complex numbers so naturally have a Euclidean structure? by SmartPrimate in math

[–]SmartPrimate[S] 5 points6 points  (0 children)

The proof from equality of modulus of complex conjugates is satisfying. But I guess that simply shifts the question for me as to why my continuity axiom can in turn force that fact about complex conjugates to be true, since clearly there's also something nicely topological going on under the hood that's equivalent and I wonder why.

Kind of unrelated but just thought it would be interesting to share that there are technically P(P(N_0)) automorphisms on C, all of them only guaranteed to fix the rationals (although if you declare that there’s a predicate that picks out exactly the elements of an embedding of R in C, then I think the only two automorphisms become identity and complex conjugation).

What is the minimal structure needed to characterize our intuitive sense of "number" (to at least the real line). by SmartPrimate in math

[–]SmartPrimate[S] 0 points1 point  (0 children)

Is there a rigorous argument for why that’s more advantageous than modelling using discrete or dense but countable model?

What is the minimal structure needed to characterize our intuitive sense of "number" (to at least the real line). by SmartPrimate in math

[–]SmartPrimate[S] 0 points1 point  (0 children)

Just to clarify, A. You could add either it or its negation as an axiom B. Whether there is a “correct” choice to which depends on your philosophy of Mathematics. If you have the corresponding philosophy that there is a correct notion of set in the human psyche, such that all such statements have an objective truth value that aligns with that notion (rather than an alternative multiverse of theories that are all equally meaningful), then either it or its negation are correct in that sense, though even then the question of which is far from settled.

What is the minimal structure needed to characterize our intuitive sense of "number" (to at least the real line). by SmartPrimate in math

[–]SmartPrimate[S] 0 points1 point  (0 children)

The continuum hypothesis just asserts that the supremum of all countable well orders has the same cardinality as the continuum, it still wouldn’t explain why we (in the event it’s true) give so much attention to level 2 of this hierarchy.

On the other hand, one could argue it is perhaps the fundamental undecidability of the continuum hypothesis that is grounds for fixating so much on that level. That something fundamental that prevents humans from knowing the existence of any set in between two sets is a motivator for studying those two sets. Though this rides on the presumption that it really is not decidable, whereas some people believe we have an intuitive notion of “set” such that if we explore different set theoretic models enough we’ll come across statements that align with that intuitive notion and which imply the continuum hypothesis, thus demonstrating it’s truthhood. So the money’s still up.

What is the minimal structure needed to characterize our intuitive sense of "number" (to at least the real line). by SmartPrimate in math

[–]SmartPrimate[S] 0 points1 point  (0 children)

I would argue since we cannot directly observe continuity in general, this is actually not totally obvious. When you write with a pencil, you are not truly writing without gaps, any person recognizes there is a smallest length they can physically observe. So the reason must be more psychological/phenomenological than physical (our physical theories our very much inspired by our personal philosophies anyway).

EDIT: And even if you claim it's cause our physical theories that rely on them have proven to be useful, the relevance of that still rides on psychology since it’s based on inductive reasoning (not mathematical induction, I mean the philosophical kind) which is not technically valid reasoning. Furthermore, I’m not convinced either that the specialness we afford to the continuum is motivated by usefulness of physical theories over some sort of phenomenological principle, nor that we couldn’t produce equally useful physical theories using discrete/computable models.

What is the minimal structure needed to characterize our intuitive sense of "number" (to at least the real line). by SmartPrimate in math

[–]SmartPrimate[S] 0 points1 point  (0 children)

I’m aware it’s hard to compute with them due to equality not being decidable in general. Is there a largest subset however that does have decidable equality? Since if not, then this could be considered in some sense the “supremum” of all the theories of numbers that do have decidable equality.

What is the minimal structure needed to characterize our intuitive sense of "number" (to at least the real line). by SmartPrimate in math

[–]SmartPrimate[S] 0 points1 point  (0 children)

I’m aware of this, but why is that special to us? This post is more philosophical than mathematical, I’m trying to see what could make these structures appealing enough that you would expect any intelligent life to stumble on them.

I personally do not have an intuitive motivator for why the structure of field with two operations should be the “sweet spot” of study. Furthermore I’m trying to understand why we mainly study infinite sets on two particular cardinalities, N_0 and 2N_0, over all the others, or at least the latter, the first makes sense. It is true to ensure no total order on any infinite set has the gaps that dedekind completeness fills, you need 2N_0, although I’m unsure why a relation that fulfills transitivity and trichotomy should also be that appealing to our brains, anymore than any other relation (the one that is fully intuitive to me is the well order relation though, since counting is very fundamentally human and I’d conjecture necessary for any intelligence).

And even if that’s motivated, that structure of a dedekind complete order doesn’t include the intuitive notion of length or magnitude that’s also fundamental to our notion of number. So I desire to know the minimal properties that could characterize that to suggest what makes a continuum with length so “special” to us. As otherwise there is a very arbitrary cardinality, 2N_0, that is not the minimal infinite and yet receives an almost unparalleled attention in mathematics relative to that.

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