#foundational mathematics

[m---a---x]

Inspired by all the newly created communities i have also created one about the topic closest to my heart: Foundational Mathematics

It is inteded for all types of posts about and from people of all kinds of backgrounds interested in the topic.

Please share with anyone you think might be interested. If you want to be added comment on this post, so I can add you.

[pv1isalsoimportant]

Mathematical platonists are so funny to me because they will say shit like "mathematics has this beautiful elegant design" meanwhile you get to the foundations and find out you have those "beautiful elegant statements" that are independent of ZF (or even ZFC), are not compatible with each other and each of them implies some insane monstrosity that shouldn't ever be true.

[boinky-doinky]

Hiiiii!! :3

It's me again and saw you have requests open. I was thinking you could draw SCP-590 (TJ) finding a stupid lookin frog.

Went from math homework to this

I hope you find joy in this silly doodle

[then-ponder]

Rate your favorite logic notation

[m---a---x]

Well, not an algebraist, but i suppose it's similar to other non-foundational pure math subjects in the sense that you kinda already know the high level. Often very intuetive stuff is very hard to formalize and effectively work with. Like, rubber sheet geometry with holes and dimensions is easy to think about but algebraic topology is hard. Calculating the area under a curve is easy, integration is hard.

So here is my own take on the state of pure mathematics and its influences on how mathematics are currently taught.

A lot of the pure-mathematical efforts in the last 150 up-until 50 years has been about how to formally express such concepts (old set theory, algebra, geometry, topology, programming, measure theory, ...) which was largely solved but very cumbersome to work with.

So the efforts since then shifted to improving that, granting us with forcing constructions, algebraic geometry, algebraic topology, functional programming, information theory, ... and then quickly expanded to making results, terminology and methodology transferable, bestowing upon us category theory, condensed mathematics, model theory, ramsey theory, ... .

Undergrad pure math mostly deals with the first kind, the formalizations, the "old" math. Resources to learn it are readily available, teachings well developed.

Gradute pure math assumes now all of that as known, dealing with technicalities has become a prerequisite and your ability to do so your justification to call yourself a mathematician. Now the focus is the current efforts with the goal of getting you to andvance them, smoothly tansitioning from teachings to research.

What neither focuses on is intuition for the problems. This has a few reasons:

For once, prerequisition. A lot of it was already obtained through school math, extra carricular interest in the subject and, well, just living your life.

Secondly, pride. Mathematics aspires to be universal and assuming it is lessens the relevance of specific problems, since everything is a relevant problem.

Thirdly, time contraints. Both in preperation, since just reading off definitions and proofs is easy, as well as in lecture hours, since everyone, allways is overly ambitious with their schedule, leading to the "non-essential" being cut.

Forth and last but very much not least, fear of corruption. And that one is definitely partly valid. Usually the examples used as intuitions do not capture the full generality and relying on them often leads to overlooking the specifics which they forget. Often those are the interesting parts, the point of it all. But reliance on intuetive alegory can get you through the early definitions, leaving one utterly lost later.

The greatest lecturers set themselfes apart in how they handle intuition. How they get you to think the right way about a subject. Not only convaying the information, but the shape of the information.

Maybe focusing on making mathematics intuetive will be the next trend in the evolution of pure mathematics. Category theory already kinda sets us up for that, as intuitions are functors and transformations.

I'd like that.

I am attempting to learn “real algebraic geometry” (schemes) and it is the most brutal. The density of new concepts is like, 5 times as high as any previous math class I’ve been in. It took us about a month to define what a scheme even is (using locally ringed spaces). I’m somewhat hoping the difficulty / density of new concepts thins out from here, but I’m not sure if it will or not. 

I do have this wondering about, learning about sheafs and categories and so on so far doesn’t feel super directly related to what schemes actually are (this is a bit silly, I suppose, since a scheme is “mostly” “just a sheaf” (plus the Zariski topology)). But I can’t help but wonder whether I’m feeling this overwhelmed because I’ve done the equivalent of learning assembly before learning C, instead of learning C or better Python and then opening it up under the hood. I do expect we’ll be using a lot of category theory and some sheaf theory though. The first big proof we’ve done so far was the proof that the global sections functor is adjoint to the Spec functor, which both is a theorem of category theory as well as using a lot of category theory in the proof. 

It is definitely the first time a math class has made me wonder if I should make flashcards to memorize things though. 

[philosophybits]

The application of the concept of ‘following a rule’ presupposes a custom. Hence it would be nonsense to say: just once in the world someone followed a rule (or a signpost; played a game, uttered a sentence, or understood one; and so on).

Ludwig Wittgenstein, Remarks on the Foundations of Mathematics

[daily-public-domain]

Day 389: Fractal Tree, 2005

link

–This image is part of the public domain, meaning you can do anything you want with it! (you could even sell it as a shirt, poster or whatever, no need to credit it!)–

[blood-orange-juice]

I wonder if the whole Abyss and Forbidden Knowledge thing is a metaphor for Gödel's incompleteness theorems.

It's not that farfetched, the second theorem was popularized in non-academic circles (to the point where even mentioning it in academic discussions is now a faux pas unless it's among mathematicians) and was used to point at the core difference between human and AI thinking (until the current generation of AI models fell outside of what it describes).

And Genshin does have a lot of AI themes.

Bear with me for a second.

In layman terms, in a system like formal logic there are statements we can express using this system and ones we can prove or disprove.

And if the system is consistent (doesn't hold contradicting statements) these are different things! You can say things that will be true and will correctly describe the object you are trying to describe but there will be no way to logically derive them from the other statements you already have.

Also a consistent system can't prove that it's consistent.

(an inconsistent, contradictory system has no such problems.

you can also formally accept a non-provable statement into a consistent system as another axiom/"known truth" and it will create a new system with new properties)

These theorems were formulated as a way to talk about the limitations of mathematics compared to our thinking, but in popular reading they somehow turned into "for any formal system there's a statement that nukes it" (Gödel statement).

Some people wondered if a Gödel statement that fries a person's brain is possible.

What I currently see in Genshin painfully resembles those popular readings.

Celestially maintained Teyvat laws are a consistent system, forbidden knowledge is a statement that can be neither proven nor disproven in that system (and messes it up), Abyss is an inconsistent system (any statement can be proven true and it's pretty horrible but also hey, anything can be proven true so forbidden knowledge doesn't disrupt it further).

Perhaps gods and their ideas are something like logical statements and owning a gnosis symbolises incorporating their "truth" into the overarching formal system of Teyvat laws.

(perhaps different cycles are just variations of the same world with slightly different rules/axioms)

Also Irminsul cannot be allowed to think of things it can't understand and it cannot be allowed to think of its nature and consistency of the information it stores.

[raffaellopalandri]

Book of the Day - SGA 4

Today’s Book of the Day is Theorie des Topos et Cohomologie Etale des Schemas. Seminaire de Geometrie Algebrique du Bois-Marie 1963-1964 (SGA 4), curated by Michael Artin, A. Grothendieck, and J.L. Verdier in 1972 and published by Springer Verlag. I have chosen this book as I used some of its math to demonstrate to a colleague some topological errors he was making in creating a cognitive…

[pv1isalsoimportant]

Mathematics is the mirror image of humanity.

[forgottenbones]

What IS a Number? As Explained by a Mathematician

[bagalois]

notes from "foundations of combinatorial topology" by lev pontryagin

french mathematician poincare was a real pioneer of combinatorial topology. he was the guy who came up with the fundamental notion of "given a n-dimensional manifold M and a sub-manifold Z, there either exists or doesn't exist a sub-manifold C that has Z as its boundary." he was also the guy who came up with the idea that manifolds can be decomposed into simpler parts, called simplices. nowadays we learn this stuff when we do homology theory, as a lead-in to algebraic topology. and, apparently, homology theory is the foundation of this stuff called combinatorial topology too.

Combinatorial topology studies geometric forms by decomposing them into the simplest geometric figures, simplexes, which adjoin one another in a regular fashion.

simplexes (and polyhedra, which are created by building simplexes together) can be examined in a group theoretic way. similar to how elementary algebra was created and makes a lot of geometry problems trivial, there are numerical invariants in simplexes/polyhedra and we can basically treat these geometric objects as just numbers. then, just by tweaking the same methods, it becomes possible to examine more complicated geometric forms which may not be reducible to numbers.

(notes from the introduction)

[then-ponder]

Truthfully, I am critical of most arguments that urge people to learn math. This is in particular because most public defenders of mathematics and math education argue with the implicit assumption that you should learn this if and only if it is useful. They will argue that because we can use all of our math to travel the stars and save our dying planet it is worth learning.

I do not blame them for this. After all this is a common practice under capitalism. The constant need to prove useful is a hellish practice in which all professions participate even if some have it easier than others. But it is harmful. It separates us from seeing math for what it is. Want it wants to be.

Many people would call math a science and there is a truth to that. Much of mathematics is a process of learning cold hard facts, even if the idea has been proven a million times over, but that ignores all the bits of it that we tend to not include. The parts about logic and reasoning, the parts about discovery, the parts that suggest a deeper explanation, the parts where intuition feels like magic and reality feels fake.

At the beginning of this post I said that “arguing math is useful is a harmful argument”. This is why. There is no discovery of facts. Logic, what should be the foundation of all mathematics, is not taught and proving that a statement is true goes ignored and is deathly fatal to students who seek the answers, yet are obscured. Worse, the facts we teach in math are plainly obvious to the student in a way that destroy any merit of the claim “ math is useful” because why would I use math when I could use logic. Why would I use the Pythagorean theorem when I know that two sides of a triangle are longer that the hypotenuse when I’m path finding and why the hell would I use a polynomial to describe anything. You use the language of usefulness you should be teaching useful things, but math can’t teach useful things. It’s math not a trade or a science.

And in truth mathematics isn’t useful. Nothing you point to that most people understand are useful things and the vast majority of research is on problems that most people either don’t care about or find utter nonsense. Truly I don’t mean this to be cruel, it’s just that the proof for 1+1 = 2 is a books worth of pages. And that’s the thing people can’t possibly be bothered to ask questions that a mathematician would lose years to. They just want a fucking answer and that’s fine, but this more explicit explanation is why Mathematicians don’t like ‘applied’ math. It’s because assholes take our prized possessions, our meow meows, and our blorbos and they throw them in a shredder and mangle them up till they have a thing that they like and then they give it to another person who fucking hates math because they got their degree in “I hate math but I love problem solving so I became a [insert stem field here]” so they can blow up that thing so that they can use it to solve something.

The truth is that math is a humanity because it is a reflection of our own reality. Quite like art. And the reason for studying art is the same reason you should study math which also happens to be the reason why math and art education sucks. You should learn about these things because you want to. No amount of money can justify it. You have to want it. And the thing is art is best taught as art. It’s not a just or simply a tool for people to make money. It’s an expression of the self - The very people who created it.

Many fear that if we allow people to stop studying the math people won’t. Fair, but high school math is not reflective of math. It doesn’t teach foundations, it doesn’t teach about the different sub-disciplines. It refuses to teach any math that can’t be directly applied to science and engineering. Of course people don’t want to learn that. You’ve ripped out the meat.

[ricisidro]

Can't wait for Season 3… 🪐

Foundation (2021) is the saga of Harry Selden and his followers on their epic quest to save mankind based on the scifi novels (1951-1953) of the same title by Isaac Asimov.

Seasons 1 and 2 with 20 episodes are now streaming on Apple Tv Plus.

#ScienceFiction #psychohistory #history #sociology #mathematics #statistics #GalacticEmpire

[the-blossica-fan]

I return with yet another 12 hours without sleep.

Pirate AU! Everyone is a pirate in the Caribbian or something, ever watch Pirates of the Caribian? Wooteloodle-loo.

I have never watched Pirates of the Caribbean 😓

I think I need someone to educate me in films because what did I even watch in my life 😭

I love the idea! I actually had a mild Pirate AU going through my brain, it didn't include everyone but I thought it would be quite fun.

...Oh my God why did I just imagine Isolde in a pirate garment. Ignore that thought.

Regulus would finally be a real pirate with a huge ship, and I like to think she gets along with Windsong and has her as the one marking the next destination, she finds use in Windsong's scholar knowledge and her ability to map. They'd be good friends.

Regulus' ship would be full of people she found interesting, like say Melania and Windsong. And Vila would join sooner or later because Regulus knows.

Vertin's ship would be a standard Foundation ship given to her and she has to end up asking for a bigger one because she has enough population to fill a small town

[education43]

What Are the Qualifications for a Data Scientist?

In today's data-driven world, the role of a data scientist has become one of the most coveted career paths. With businesses relying on data for decision-making, understanding customer behavior, and improving products, the demand for skilled professionals who can analyze, interpret, and extract value from data is at an all-time high. If you're wondering what qualifications are needed to become a successful data scientist, how DataCouncil can help you get there, and why a data science course in Pune is a great option, this blog has the answers.

The Key Qualifications for a Data Scientist

To succeed as a data scientist, a mix of technical skills, education, and hands-on experience is essential. Here are the core qualifications required:

1. Educational Background

A strong foundation in mathematics, statistics, or computer science is typically expected. Most data scientists hold at least a bachelor’s degree in one of these fields, with many pursuing higher education such as a master's or a Ph.D. A data science course in Pune with DataCouncil can bridge this gap, offering the academic and practical knowledge required for a strong start in the industry.

2. Proficiency in Programming Languages

Programming is at the heart of data science. You need to be comfortable with languages like Python, R, and SQL, which are widely used for data analysis, machine learning, and database management. A comprehensive data science course in Pune will teach these programming skills from scratch, ensuring you become proficient in coding for data science tasks.

3. Understanding of Machine Learning

Data scientists must have a solid grasp of machine learning techniques and algorithms such as regression, clustering, and decision trees. By enrolling in a DataCouncil course, you'll learn how to implement machine learning models to analyze data and make predictions, an essential qualification for landing a data science job.

4. Data Wrangling Skills

Raw data is often messy and unstructured, and a good data scientist needs to be adept at cleaning and processing data before it can be analyzed. DataCouncil's data science course in Pune includes practical training in tools like Pandas and Numpy for effective data wrangling, helping you develop a strong skill set in this critical area.

5. Statistical Knowledge

Statistical analysis forms the backbone of data science. Knowledge of probability, hypothesis testing, and statistical modeling allows data scientists to draw meaningful insights from data. A structured data science course in Pune offers the theoretical and practical aspects of statistics required to excel.

6. Communication and Data Visualization Skills

Being able to explain your findings in a clear and concise manner is crucial. Data scientists often need to communicate with non-technical stakeholders, making tools like Tableau, Power BI, and Matplotlib essential for creating insightful visualizations. DataCouncil’s data science course in Pune includes modules on data visualization, which can help you present data in a way that’s easy to understand.

7. Domain Knowledge

Apart from technical skills, understanding the industry you work in is a major asset. Whether it’s healthcare, finance, or e-commerce, knowing how data applies within your industry will set you apart from the competition. DataCouncil's data science course in Pune is designed to offer case studies from multiple industries, helping students gain domain-specific insights.

Why Choose DataCouncil for a Data Science Course in Pune?

If you're looking to build a successful career as a data scientist, enrolling in a data science course in Pune with DataCouncil can be your first step toward reaching your goals. Here’s why DataCouncil is the ideal choice:

Comprehensive Curriculum: The course covers everything from the basics of data science to advanced machine learning techniques.

Hands-On Projects: You'll work on real-world projects that mimic the challenges faced by data scientists in various industries.

Experienced Faculty: Learn from industry professionals who have years of experience in data science and analytics.

100% Placement Support: DataCouncil provides job assistance to help you land a data science job in Pune or anywhere else, making it a great investment in your future.

Flexible Learning Options: With both weekday and weekend batches, DataCouncil ensures that you can learn at your own pace without compromising your current commitments.

Conclusion

Becoming a data scientist requires a combination of technical expertise, analytical skills, and industry knowledge. By enrolling in a data science course in Pune with DataCouncil, you can gain all the qualifications you need to thrive in this exciting field. Whether you're a fresher looking to start your career or a professional wanting to upskill, this course will equip you with the knowledge, skills, and practical experience to succeed as a data scientist.

Explore DataCouncil’s offerings today and take the first step toward unlocking a rewarding career in data science! Looking for the best data science course in Pune? DataCouncil offers comprehensive data science classes in Pune, designed to equip you with the skills to excel in this booming field. Our data science course in Pune covers everything from data analysis to machine learning, with competitive data science course fees in Pune. We provide job-oriented programs, making us the best institute for data science in Pune with placement support. Explore online data science training in Pune and take your career to new heights!