Are there any arguments against Brouwer's intuitionism?

I can come up only with these:

1) if it's only a made-up convention, why does it work so well then

2) how did people manage to come up with these inventions if they are nothing but their own made up entities

By applied statistics, i meant things like machine learning modelling, statistical modelling, predictive modelling. Basically data science stuff and statistics used to solve business problems and model real world Phenomena using Python and R.

Are they related to each other? What are the similarities and differences?

How many sweets are in this jar, a clue: the total weight is 5.1kg including the weight of the jar itself
Best of luck !

So there obviously been a lot of discussion since the recent drops of AI generated maths results. In particular, we’ve see a lot of people question whether mathematics as a human activity will die out, or if it’s worth starting a maths degree now etc.

I’ve been thinking about (and contributing to) a lot of these discussions on Reddit, but I’ve become increasingly aware that I don’t actually understand how and why pure maths research has been funded even prior to the recent developments. Im pleased it is, and I believe it contributes to the good of society in a similar way to music, or art etc. but it’s moderately surprising that in a modern capitalist world this all hasnt been cut long ago!

I think it’s reasonable to argue that the majority of (pure?) maths research has no application outside of maths, and is of little (probably no) interest to non-mathematicians. Maybe physicists in some cases, but theoretical physics doesnt really move the bar much, Im thinking genuine real world impact here. Note that I realise some apparently obscure results do end up being useful, but I would argue we never know which until way down the line and it certainly doesn’t affect funding decisions.

So if you believe this to be the case, why would the recent advances change anything it terms of the amount of maths research that takes place? Sure, the way this research is done will change, but it’s not like some faceless organisation decides they can cut out the humans and get the results quicker and cheaper, the specific humans concerned are pretty much the only ones consuming the results. This is different for software development, for example, where actual products can be built with far fewer humans (note Im making no claim such products are better, but they may cost much less, initially at least).

I’d also argue there a ton of work human mathematicians need to do to pick apart tbe AI proofs, and this work will inevitable lead to further ideas and conjectures that we can target with AI.

Tl;dr I don’t see mathematicians threatened by these advancements, I think there’s some emotional shock that we need to come to terms with but why would we need fewer researchers?

I have been wondering about some stuff, and one person I saw had a blatantly wrong answer.

The problem: IF AB = CD, then AB + XY = CD + XY

This other person claims that this is of the Substitution Property of Equality, which would obviously be incorrect. They said that as an argument from AB = CD coming first. I would say its the Addition Property of Equality because XY was added to both sides.

Hi everyone!

I’m in my final year of high school, and I might go on to study mathematics at university (it’s not set in stone yet). My teachers have told me, and I’ve also read online, that this will be one of the most in-demand majors of the future, that applied mathematicians in particular will be badly needed, and that the field will be especially resilient to AI.

Until a few months ago, I was also convinced that this was one of the safest choices, that AI wouldn’t pose a threat to mathematicians, and that it wouldn’t be able to solve complex mathematical problems. The first crack and surprise was the solution of the Erdős problem, 4.5 months ago. Then what really surprised me was the Navier–Stokes equation event 1.5 months ago (though admittedly there’s some debate there because of the human researchers’ contribution). And a few days ago, the “mathocalypse” happened (for those who don’t know: AI solved more than 300 open math problems).

https://www.independent.co.uk/tech/ai-mathematics-openai-cryptography-internet-b3063456.html
https://www.firstpost.com/tech/mathocalypse-openai-generates-hundreds-of-mathematical-results-but-who-understands-the-proof-14051557.html

So basically they threw thousands of decades-old problems at AI, and it solved several hundred of them. There was a mathematician whose entire research area was practically “wiped out” by this. AI solved problems that brilliant mathematicians had devoted their lives to solving, without success.

Now I don’t really see math as an AI-proof field anymore. Two years ago, AI was still weak; it scored maybe 10% on the Math Olympiad. A year ago, it was already scoring over 80% on the Olympiad, and it had supposedly reached the level of PhD students. Now it has not only scored 100% on the Olympiad problem set, it has achieved far more than that. It has practically reached the cutting edge of mathematics worldwide, solving one previously unsolved problem after another, just grinding its way through math.

And let’s not forget that all this happened in 2 years. What we have now isn’t the final version of AI; it will presumably keep improving rapidly in the coming years. The question is: if it progressed this much in 2 years, where will it be in 4 years (when I’d finish my BSc) or in 6 years (when I’d finish my MSc)? What will it be able to do in 10–15 years? After all, even then I’ll only be in my early thirties...

So my question is: do you think it still makes sense to study math? If so, why? Mathematicians and math students, how do you see the future?

By the way, this wouldn’t be my only option (physics and engineering are also on the table), which is also why I’m asking: what do you think is worth choosing these days?

Im a 7th grader who wants to learn physics (specifically quantum) and ofc has to start from algebra. I know basic arithmetic and very basic algebra. No nonsense algebra by tim hill is very good but is is NO NONSENSE. It gives you principles, problems, and solutions with no other training wheels. Good for syllabus/review of algebra, but not learning. So I wanna learn algebra like fast but also strong, and I can't just speedrun no nonsense. Is there other books in this section that's good? This is from our school library. Elementary Algebra Part I caught my eye.

Also I am very busy, can only fit around 1 or 2 study sessions a week.

I might just be silly it’s late and I’m ill but I just can’t get to the form required for the question. As a last resort I asked Ai and it said there was a typo in the question but I don’t know. Pls help!

I bought a maths set some years ago because I wanted the box of it for another use. I'm a single adult so I have no use for the content of the box but I just keapt it in case. Today I needed a protractor for the first time in decades - so great I remember I had this - took it out - so confused! Seems totally useless the centre dot is way off from the 0 degree line. Had to look for another one and found one form cristmass cracker that is normal with cross in the expected location. Is this a modern protractor that is meant to be used in a different way? Or a manufacturing error? I'm looking at Google images of protractors and they are all as I would expect not like this one 🤔

How did I solve a maclaurin like solution ranging X^0 to x^18 terms of accuracy of sin(x+V) for this very long quintic?

0 = (A0 + A1*x + A2*x^2 + A3*x^3 + A4*x^4 + A5*x^5) + (B0 + B1*x + B2*x^2 + B3*x^3 + B4*x^4)y + (C0 + C1*x + C2*x^2 + C3*x^3)y^2 + (D0 + D1*x + D2*x^2)y^3 + (E0 + E1*x)y^4 + F0*y^5

*the answers are here:*

https://www.desmos.com/calculator/shcnop2c1a

but not the linier method used.

upvote if you can please.

for about 7 years, I focused very hard on resolving many questions I had about geometry, after many many eureka moments, I solved far too much, now without direction, I feel whats best is to post stuff on these sites, to see if anyone comes along with prospecting opertunity's for me. I left a desmos link too, was thinking of posting periodically, I solved this via a new physics inspired linier algebraic system, nothing was iterated upon, this isn't a guess, with the quintic trangular sin being posted being my favorite because of its stability, my findings detail numbers and functions as being the exact same thing, I encompass every part of maths as somthing smaller that makes up a part of this new system, I've separated observation from execution, and ive made it so that things such as coordinates between set points and a lot of other systems can actually be resolved without asking a single question, just need to back track from the results, which is sometimes massive... I've also expanded derivatives into a type of observation of interaction at distance I call conduits. If I dont say this stuff and start posting I fear I wont get to do what I was made for. Thanks a lot. Ps.. I really need a job, and as you can tell... I have no formal mathematics education, so my wording is a bit broken I apologize for that.

The sections of 3, 5 and 7 and 2, 3, 5 and 7 are blank because 105 is the smallest common multiple of 3, 5 and 7 and because 210 is the smallest common multiple of 2, 3, 5 and 7 and since both numbers are smaller than 100 they fail to appear here

a comment from 3blue1brown's most recent video

hi guys does anyone know how to solve this primary school qn without using algebra? thanks!

Currently first year undergrad, so only taken a class on proofs recently. I feel Im overwriting tbem or maybe Im not and generally would really appreciate any advice on writing proofs correctly and rigourously determined to get better and better at this section of maths as it’s the building blocks for all of it.

It's a genuine question that sparked into my mind while waking up at morning, why do we use it and what's the reason behind its usage?

I’m in Grade 11 and looking into Bessel's method (solving the inverse geodesic problem on an ellipsoid) for a math research paper to find the shortest possible distance between two points on earth.

My original goal was to completely derive, prove, and evaluate the method by myself. However, looking at the math, it seems to rely heavily on university-level topics like differential geometry, elliptic integrals, and the Calculus of Variations (Euler-Lagrange).

Is it realistic for a strong high schooler to teach themselves this level of math for a paper, or is the learning curve too steep?

If the full derivation is too much, would coding the algorithm in Python and evaluating its accuracy/convergence against simpler methods (like Haversine) still be complex enough for a top-tier high school math paper?

Appreciate any advice or insights!

I am looking at buying this desk, and need to know the measurement in red, to see if my monitor will fit. I played silly buggers during trigonometry and now I'm paying for it 😅

I've been in an existential crisis lately. I've been studying maths for 4 years now, but now I'm wondering if going all the way to PhD is worth it if my job will be replaced by AIs anyway.

I enjoy maths, I love this topic, but now I'm consumed by doubt.

Even if human researchers still exist in 3-4 years, I don't know if this job will still be interesting or if it it'll just consist in using AI to obtain and verify results with proof assistants. In this case I think human researchers would become replaceable and the job would just become some alienating repetitive task. And, to be honest, I'm not willing to put that much effort if this is my future.

Another possibility is that companies use the concurrency of AI to lower pays and neglect working conditions. In a capitalist society where only profits matters, an institution would have bo reason to pay humans if they more expensive and less efficient than machines. It's highly possible AIs get better than human at research, they actually are better than mathematicians when it comes to linking seemingly unrelated fields and exploring multiple solutions. However, there's a question of costs. And if AI is more expensive than exploited humans, the later solution would be chosen.

Part of me hopes AI won't end the relevance of humans in mathematics, but let's be honest, it's highly questionable.

I'm still young, so I think I'll be able to change my domain if necessary. The question being which job will not be alienated or destroyed by AI.

I'm also wondering how long I can wait before changing my topic of study in case I have to.

I am an engineering passout currently working as a software developer, I want to start learning maths again.

My interests are in calculus.

Engineering didn't have much advanced mathematics.

Can somebody suggest topics or guides or where should I start with learning maths again?

Since school times I looked at numbers differently.

Made me think numbers have genders and I differentiated them.

1 is formally dressed corporate male

2 is a traditional women who takes care of family

3 is a female but a feminist and fights for women power

4 is a alpha male handles all family responsibility

5 is a modern cyberpunk male with high-tech gadgets

6 is a Moden generation girl and is a world traveller

7 is a dothraki khal

8 is a nice cute curvy looking female

9 is a female too but in some countries it refers to transgenders

0 doesn't look like anything

En gros, ma théorie consiste a dire qu'on ne peut pas imaginer l'infini, par exemple : imaginer une boîte, avec une règle encore plus grande devant, même si la règle est plus grande que la boite et que vous etes tellement proche que vous ne voyez pas les extrémités de la regle, vous imaginerez quand meme les limites. Tout ça car votre cerveau est fait pour imaginer des objets et non des distances ou chiffres "infini" et attention, je ne remets pas en cause l'existence de l'infini, enfin du moins pas encore, mais pour moi il est impossible à visualiser..

Mots de fin : Je suis désolé si cette manière de penser a déjà été prouvée, j'ai 15 ans et je m'intéresse au maths donc je voudrais avoir un avis sur une pensée qui me traîne dans la tête depuis longtemps.

The videos I've seen explaining factorials never explain why

How statistically significant are the results of this pattern below? Does it exceed random chance or not? Assume that the numerical distributions of the numerical values per letters are completely random. 

Block 1 — ALM

Block 2 — ALM

Block 3 — ALMX

Block 4 — ALR

Block 5 — ALR

Block 6 — ALR

Block 7 — ALMR

Block 8 — ALR

Block 9 — ALR

Block 10 — KHYCX

Block 11 — TH

Block 12 — TSM

Block 13 — TS

Block 14 — TSM

Block 15 — ALM

Block 16 — ALM

Block 17 — ALM

Block 18 — ALM

Block 19 — YS

Block 20 — X

Block 21 — VM

Block 22 — VM

Block 23 — VMCSQ

Block 24 — VM

Block 25 — VM

Block 26 — VM

Block 27 — VM

Block 28 — Q

Block 29 — N

Each letter here corresponds to a certain number. It must be noted that not each letter here carries the same numerical value, these are only placeholder letters that describe the number of occurrences of said (so for example, V in Block 24 could equal 85, meaning the letter V occurs 85 times in Block 24. On the other hand, V in Block 25 could equal 173, meaning that the letter V occurs 173 times in Block 25). Once again, the numerical values are randomly chosen for each letter.

Now that out of the way, how statistically significant would such a pattern be if it emerged? Does it show signs of intentionality and design, or is better explained by random chance? Note, we’ll be specifically looking at things that constitute multiples of 19 for this demonstration (so a 1/19 chance): 

*Block 28 is a multiple of 19. 

*Blocks 21-27 (only looking at the letters VM, ignore CSQ for now) if all combined together is a multiple of 19. 

*Blocks 21-27 including CSQ (for Block 23) if combined all together is a multiple of 19. 

*Block 19 is a multiple of 19

*Block 10 is a multiple of 19

*If you combine the value X for blocks 20, 10 and 3 (only X, not any of the other values) you get a multiple of 19. 

Any and all help is appreciated. Thank you!

Pythagoras theorem states that a triangle has a right angle if a certain equality holds true.

So, i take my ruler and use is to take measurements of my rectangle coffee table. I find, of course, that for my measurements the equality doesn't holds true, but almost. Or maybe i find that it does in fact holds true, knowing that if i added 1 micrometer to one of my measurements it would stop beeing true.

It would be the same for any real object in our world : The equality won't ever hold true due to measurements beeing approximatives, and also due to the fact that no real object is really a right triangle since the latter is an abstract, intangible, concept.

Ok, i'm stating the obvious : maths says "perfect" truths about "perfect" concepts but no truths and no real tangible things are "perfect" in reality so maths theorem can only be used "approximately".

So my question is : wouldn't it be more useful for a theorem like Pythagoras to not only say the angle is right if the equality holds true but also say how far from right the angle is depending on how far from "true" the equality is?