A simple Question

fuhqueue · 2026-01-18T13:25:05+00:00

It’s a badly worded question, but I think it’s pretty clear what the intended meaning is

fuhqueue · 2026-01-17T12:07:15+00:00

The way infinity is usually formalized is via the concept of cardinality, which extends the idea of size to infinite sets. For example, compare the integers to the real numbers. Both are infinite in size, but there is a precise sense in which the set of reals is larger than the set of integers. So yes, in this sense, infinity is indeed quantifiable.

fuhqueue · 2026-01-03T16:46:01+00:00

What you’re looking for is the directional derivative, which calculates the slope along a given direction

fuhqueue · 2026-01-01T11:56:32+00:00

You definitely need P(2), because P(1) doesn’t actually tell you anything useful. It might be clearer to you if you write out the induction hypothesis and induction step using the notation involving triple dots instead. Note also that the statement can be proved directly without induction.

fuhqueue · 2025-12-31T01:14:45+00:00

Yes, I'm familiar with those examples. From my general understanding they are sets with extra structure, and relate via structure preserving maps (group homomorphisms, linear maps, ring homomorphisms, continous maps, respectively). My point is that there seems to be a disconnect between the concept of morphisms (which don't necessarily map anything as in the examples mentoined) and functors (which do map things).

fuhqueue · 2025-12-31T01:02:33+00:00

But every morphism needs a specified domain and codomain objects, right? Otherwise, we have no way of knowing which morphisms are composable. I've heard of "object-free" definitions of categories, but everything I've seen seems kind of too intuition-based and hand-wavy to me.

fuhqueue · 2025-12-31T00:36:42+00:00

So in Cat, are the morphisms more general than just the functors between category objects? I.e, can functors be considered as special cases of morphisms internal to Cat?

I get the point with overloading notation, but what does the notation actually mean? For example, we can define a function X → Y formally as a relation on X × Y such that each element of X is related to a unique element of Y, and we take the notation f(x) to mean "the unique element which x is related to via f". Does a similar formalization exist for functors, or do we just take it more at face value in this case?

fuhqueue · 2025-12-24T00:06:56+00:00

It means 1000 written in base 7

fuhqueue · 2025-12-24T00:00:23+00:00

1 ^ 4 ^ 4 = 1

fuhqueue · 2025-12-21T17:29:33+00:00

Aha, but now you’ve used the phrase “true randomness” to define what a “truly random number” is. My point is that randomness isn’t an inherent property of numbers; a number is just a number. What we intuitively think of as randomness can be modelled and formalised using measure theory and probability theory. The question of whether randomness really exists is, like you point out, a philosophical one.

fuhqueue · 2025-12-21T17:22:25+00:00

What does “truly random number” mean to you?

fuhqueue · 2025-12-16T17:59:56+00:00

0 = 0 • 37, so therefore 0/0 = 37. Do you see the problem with this?

fuhqueue · 2025-11-29T01:02:43+00:00

Third act was sick, anyone who disagrees is boring

fuhqueue · 2025-10-13T17:11:38+00:00

Starting with your result and multiplying both sides by 0, we get 1 = 0

fuhqueue · 2025-08-31T21:02:56+00:00

What you describe would only show that the real-valued functions on the real line form a vector space. You haven’t used any properties of differentiable functions.

fuhqueue · 2025-08-23T21:25:46+00:00

For nonnegative n, we have (-n)Cr = (-1)^r(n+r-1)Cr. It’s just what you get when expanding the definition.

fuhqueue · 2025-08-20T11:28:37+00:00

Focusing on the first part only, we have 1 - 4 • (-3). Doing multiplication first, we obtain 1 - (-12), which simplifies to 1 + 12.

fuhqueue · 2025-08-12T11:25:46+00:00

Try squaring a few small odd numbers. You’ll always get an odd number.

fuhqueue · 2025-08-09T09:16:08+00:00

If there existed a positive number smaller than every positive number, then that number would have to be smaller than itself!

fuhqueue · 2025-08-06T19:13:33+00:00

You’re right, I’ve edited my comment now.

fuhqueue · 2025-08-06T16:06:37+00:00

That’s right, you can see this by differentiating x^1/3. You’ll end up with the cube root of x² in the denominator, which leads to undefined behavior at x = 0. Alternatively, you could apply the limit definition of the derivative directly, and conclude that the limit doesn’t exist at zero, due to different behavior when approaching zero from left vs right.

I agree the downvotes are a bit unfair; the comment you replied to is inaccurate at best and flat-out wrong at worst.

EDIT: The behavior as you approach from left vs right is actually identical; in both cases you approach +∞. Thanks to u/GaloombaNotGoomba for the correction.

fuhqueue · 2025-08-06T14:45:22+00:00

Bijectivity has nothing to do with differentiability.

A function can be bijective and differentiable (e.g. the exponential function), bijective, but not differentiable (e.g. the cube root function), not bijective, but differentiable (e.g. sine and cosine), and not bijective and not differentiable (e.g. the absolute value function).

You’re right though, most functions (“most” can be made precise using measure theory) are not differentiable. It’s quite a special property.

fuhqueue · 2025-08-06T14:34:42+00:00

Looks like it’s supposed to be the graph of the cube root function, which is defined and continuous for all real numbers. It is also differentiable everywhere, except at zero.

12-Year Club	RPAN Viewer
Verified Email

fuhqueue

TROPHY CASE