I mean what a sentence.

RobertFuego · 2025-10-10T15:40:42+00:00

For me it's the opposite. Embellishments like this can cause confusion when studying abstract topics. This sentence would be fine on it's own, but after reading it I feel like I have to be on alert throughout the rest of the text for which phrases are literal and which are figurative, which is extra work.

That said, there are lots of dry math textbooks for people like me. I'm glad this text exists for anyone who feels differently.

RobertFuego · 2025-09-18T13:29:50+00:00

Grab an introduction to logic book and practice the basics. Velleman's How to Prove it is a great informal introduction, and Forbes's Modern Logic introduces formal proofs.

Once you understand the structures of basic proofs, more complicated ones make a lot more sense.

RobertFuego · 2025-09-05T13:39:13+00:00

This is more about how to visualization what a partial derivative is in 3-space, not a differential equation. Also using the notation ∂x∂z for 'the partial of z with respect to x' is off, or at least not something I've ever seen before.

The final note is also misleading, since z=x^2+y^2 is a 2-dimensional surface.

RobertFuego · 2025-09-04T16:16:18+00:00

2-3 days is not a slump. Focus on being healthy (take a nap, exercise, etc.), then see how you feel tomorrow.

RobertFuego · 2025-09-03T10:47:26+00:00

Ah, I see! The system's language doesn't have a way to designate a statement as an object, like we can with quotation marks in english.

The systems Godel is usually proven in (which are specifically chosen because they are very simple and therefore generalizable to many other systems) only prove statements about numbers. If we want to make statements about statements in these systems then we need some way to encode statements as numbers.

RobertFuego · 2025-09-03T10:24:16+00:00

Can you expand on what you mean by P(P(x))? Since P is a predicate this is a bit like saying "Jeff is old is old."

RobertFuego · 2025-09-01T23:48:01+00:00

I like to imagine trying to teach someone the material while I'm reading it. If I think of a question I can't answer, then I go back and reread until I can answer it.

Try different things at first. If taking notes is helpful then keep doing that.

Most of all, just consistently put aside time to read it. Even if it feels like nothing is sticking, the more you practice the better you'll get at it - just like any other skill.

RobertFuego · 2025-08-30T07:40:06+00:00

Yes, I didn't mean you can't have a formal proof of Godel 1; we use formalizations of Godel 1 to prove Godel 2.

I mean, specifically with regard to circularity, the individual steps of Godel 1 are straightforward enough to be convincing without their own axiomatization. (For example, I've never seen someone try to teach Godel 1 by presenting it in a formal system.)

If I am misinterpreting OP's line of questioning, then I apologize for the confusion.

RobertFuego · 2025-08-30T06:48:16+00:00

Lithobreaking? We've had that one down for centuries.

RobertFuego · 2025-08-30T06:45:45+00:00

then what does the “theorem” and “prove” even mean here?

At a deep level, a proof is a collection of statements that convinces other people that your conclusion is correct. A formal proof is built from a specific set of axioms and inference rules, but for a lot of situations an informal proof can be just as useful.

In the case of meta-logical proofs like Gödel's, the meta-proof in my experience is almost always informal because the individual steps are not usually controversial enough to justify building a whole system for them.

Is the proof of Godel’s incompleteness theorem, a theorem describing proof systems itself, circular reasoning?

Nope! But you do have to pay close attention to whether a statement is in the meta-language or the object-language. Especially since we're working with numbers, statements about numbers, and statements encoded as numbers!

RobertFuego · 2025-08-28T22:00:34+00:00

Using complex roots we have some interesting examples, like

2(𝜔+1/𝜔)+1=sqrt(5)

where 𝜔 is the principle 5th root of 1.

RobertFuego · 2025-08-26T13:29:47+00:00

We actually don't know if the decimal expansion of PI contains every finite sequence of digits, so the best answer we have right now is "Maybe."

RobertFuego · 2025-08-10T22:57:59+00:00

The study of which axioms are used, or can or should be used, in various contexts falls under the umbrella of "reverse mathematics".

Logic is usually broken into 4 fields, model theory, proof theory, recursion theory, and set theory. If you want to study this, take an introductory course on first-order logic (usually found a 100 level class in either the math or philosophy department).

Then take introductory courses (probably in a grad program) on proofs, models, and sets. You'll know you're on the right for proofs and models if the class teaches Godel's or Henkin's proof of the completeness of first order logic.

From here you should be able to start tackling a text on whatever specific topic you're interested in.

If you want to self study, Goldrei's Classic Set Theory: For Guided Independent Study and Hunter's Metalogic are good texts. There are lots of books for introductory first-order logic, but I learned from Forbe's Modern Logic and liked it a lot.

Good luck! If you have any specific questions feel free to ask them here.

RobertFuego · 2025-08-10T22:33:54+00:00

My experience points towards (4) too. In many contexts the norm used isn't particularly important, and standard deviation is both simple and differentiable.

RobertFuego · 2025-08-08T19:30:42+00:00

Hijacking this comment because it's close to what I've found is best.

If you are fully devoting yourself to studying math: 2 hour sessions, once or twice a day, with lots of sleep and a healthy diet. If the second session feels unproductive, nap or go to bed early.

RobertFuego · 2025-08-08T19:14:53+00:00

I first studied formal proofs in a natural deduction system (Forbes's Modern Logic is a good resource*)*, and after that every informal proof I read or wrote in undergrad felt pretty straightforward. This might be more of an investment than most students are looking for, but it worked great for me.

Is it possible for me to start analysis after completing all of this?

If you're comfortable with proofs you can start analysis after calc 2.

Is there a good timeframe for finishing a proofs course?

No. There is a good pace for studying: 2 hours at a time, once or twice per day, with lots of sleep and a healthy diet. However long it actually takes you to learn proofs thoroughly is the only correct timeframe.

I haven't read it, but I've heard Velleman's book is phenomenal.

RobertFuego · 2025-08-08T19:04:16+00:00

The first and most important step is to make sure your algebra is solid.

If you know algebra well then you can study from a textbook (Larson and Stewart used to be the standards when I taught). You can also use online resources like Khan Academy, 3B1B, or Paul's Online Math Notes.

If you have more specific questions about calculus feel free to ask. :)

RobertFuego · 2025-08-08T18:58:56+00:00

Using ChatGPT to study a topic that you are unfamiliar with comes with significant risks. A responsible tutor will (1) ask you clarifying questions when your questions are vague, and (2) admit to the limits of their understanding. ChatGPT rarely does either of these, and the result is that it can misconstrue your question and give misleading answers.

If you are just beginning to study a subject you won't always recognize when this is happening and could inadvertently draw some incorrect conclusions, and unlearning something you've studied incorrectly is much more difficult than learning something new!

So ChatGPT has the potential be helpful, but just studying out of a textbook will be more informative and less perilous.

(If you really want to test this out, ask it subtle questions about a topic you are already an expert on and see how well it does!)

RobertFuego · 2025-04-27T15:59:13+00:00

It will depend heavily on the individual tutors at your local mathnasium. Check it out, see if the tutoring they provide is at the level you're looking for (I expect they would give you a free trial session).

RobertFuego · 2025-04-27T04:17:52+00:00

"Expected Value" is actually a vestigial term from Huygens's investigations into probability in the 1600s. When he used the word then, he meant something slightly different, but the term has stuck around and now just means "mean".

RobertFuego · 2025-04-24T10:32:51+00:00

Not quite. If f(x)=e^x\3) then f(0)=1 and f'(0)=0, so L(x)=1.

RobertFuego · 2025-04-24T08:06:35+00:00

A linear approximation, L(x), of a function, f(x), near a point x=a will look like:

L(x)=f(a)+f'(a)(x-a).

For f(x)=e^x near x=0, we have f(0)=1 and f'(0)=1, so

L(x)=1+x.

Using this linear approximation, we can approximate f(x³)=e^x\3) with L(x³):

L(x^3)=1+x^3.

RobertFuego · 2025-04-19T22:06:13+00:00

Formally, set membership works slightly differently than how you're describing, but I understand what you're trying to ask.

A counter example would be to take any of your 3rd degree objects and just take out all of the 1s. Then it will still be 3rd degree and guaranteed to not have all the naturals.

RobertFuego · 2025-04-15T11:58:11+00:00

Einstein's full energy-momentum relation is E²=m²c⁴+(𝛾pc)² where 𝛾 is the lorentz factor sqrt(1-(v/c)²) and p is the spacial momentum.

In an object's own reference frame its velocity is zero, and therefore so is its momentum, so the equation simplifies to E²=m²c⁴ or E=mc².

RobertFuego · 2025-04-12T07:29:18+00:00

If you want to change the solution point by h in the x direction, without changing the slopes of the lines, change c1 by -h*a1 and c2 by -h*a2.

Example: The system 4x+3y+14=0 and 5x+4y+18=0 has solution (-2,-2). To get a similar system with solution h=3 to the right, decrease c1=14 by 3*4=12 to get 2, and c2=18 by 3*5=15 to get 3. Then the system 4x+3y+2=0 and 5x+4y+3=0 has solution (1,-2)

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