This is not a critique of 18th century mathematics, but mathematics beginning with the schizophrenic - George Cantor. Just like how madness corrupts rationality, set theory has corrupted logic. Rewriting the previous era’s insights with ugly single letter variables and naming concepts after themselves - the summary of last 150 years of modern mathematics. Thankfully in another 50 years all of them will be dead.
Ever since the dawn of Set theory from the Schizophrenic loon George Cantor maths is being rewritten into “pure” and “applied” maths. The theologian sure love to use the word infinite. Pure mathematicians really believe that universe can be described by set theory and set theory explains why God wants your foreskin. Are there infinite things without mass ? If not the foreskin God then which God are they talking about ? Because under the sublime and fine traditions of Human Sacrifice in Abrahamic faiths you will be put to death if you are talking about the wrong God. The theologians have yet to prove why the “humble” watch making God is the same as the Demonic God who wants endless human sacrifice and why God presents himself like a ghost of Caesar. Needless to say “pure” mathematicians have a God complex or a servant of God complex.
Pure mathematics is an organised religion and has more in common with ugly priestly languages like arabic and latin. Unicorns that don’t exist in reality and yet all books describe them. Pure mathematicians saying they can predict what physics is like a film critic saying all movies end. Russel, Whitehead, Hardy and Hilbert are fantasy writers. The predictions of pure mathematics are vaguer and less entertaining than astrologers. It’s funny how things are always going to go good or bad but never meh. Even if you are an atheist if you believe X is pure and Y is impure you produce irrational spasms just like a religion fanatic undergoing faith healing.
If you love mathematics go back to Newton and Euler and forget formal mathematics. Much to my surprise their works are readable and use simple language as opposed to the endless symbolic exorcisms of modern mathematics. Until recently maths was seen as the branch of physics. Newton, Archimedes, Euclid, Euler, Alan Turing, Von Neumann … all were physicists or engineers. They used mathematics just like how programmers use test cases. I believe what Tests are to Code, Mathematics is to Engineering and Physics. Mathematics like test cases in programming helps in making things more coherent and covers unseen edge cases. Maths is notation for this style of thinking. Seeing maths as the foundation of anything is delusional. We infer reality by experiments. Any ideas of reality which are not tested empirically are nothing but fairy tales. Maths is not science. Maths far from being the queen of sciences is its midwife. After Cantor, Mathematicians began to taken themselves way too seriously thinking they could write one proof that could prove all proofs and that they could prove mathematics with mathematics. The halting problem, Wittgenstein’s attacks, NP vs P, Godel’s theorem put an end to that bullshit.
Even Lisp programmers think the universe is written in Lisp. The insistence that sets are the center of everything, is like saying everything is made of the five elements air, water, earth, fire, ether. Yet one is a superstition and the other is a serious claim. This sorta claims are also made by OO, Lisp and Haskell programmers namely objects, lists, types are center of everything. OO, Lisp and Haskell programmers practice Software Homeopathy.
Everything in reality can be decomposed into recursive functions is a false epistemological claim. Functional programmers think the world looks like this,
In fact the entirety pure mathematics, category theory and russels paradox have been defeated by multi valued, fuzzy and even quantum logic. Reality uses multi valued logic not cantor’s sets. The law of contradiction is superstition. The advances in recent A.I are because of using Fuzzy Logic like thresholds in data analysis, not pretending that lists are the center of everything. Given what we know now, one has to marvel at the bullshit Lisp programmers have pulled for the last 60 years. With 1Hz processor and less than 1Kb RAM they claimed they could do everything in A.I and yet even in 2020 all the Lisp programmers put together can’t even write a grammar checker. I suppose NP problems have way of putting these pure programmers to shame.
To model modern day electricity you need multi valued logic. You need an “undefined” voltage value along with 0 / 1. If pure mathematicians want to insist on the law of excluded middle, I would suggest that they first refrain from using all technology just like the amish. What use is category theory if it can’t even model electricity ?
Why should programmers even use mathematical jargon ? Godel’s incompleteness theorem shows that maths can’t have any foundations, so how can category theory be the foundation of anything other than another layer of delusion ? Simulations are better than proofs. Computer science is the study of building processes. To make the process building reliable is the task of software engineering. Process is to computers what engine is to the machines. Functions are just dictionaries in disguise where the values are defined by a formula. In a sense python’s lambdas are the closest to what functions in mathematics represent … thats how limited mathematical functions are. Functions are never going to be subroutines. Functions are never going to be processes no matter how many “lambda papers” Guy Steele will write. Computer science is not a branch of mathematics. It might as well be that Actor model is what differentiates programming from mathematics and all other sciences. It seems Carl Hewitt is largely treated as a pariah for his contributions by the mathematics community. Bunch of pussies who can’t debate rationally but bitch and moan about every imaginary insult to their “sacred” dogmas. You might think we got rid of relgigion but it sneaked in with the backdoor of purity.
Haskell and Lisp will never have “pure” compilers. They will always be parasites onto “real” stack machines. Immutability is waste of memory. Pure maths is the java enterprise of mathematics. They are really just designing interfaces and want everyone to respect them as though they have discovered a cure for cancer. Java programmers thought they discovered the calculus of enterprises. Naturally it did not work when they tried it. Lisp programmers claimed they could do A.I with linked lists, 1Hz processor and 1kb RAM. Now haskell programmers promise us multicore scalability. I guess we can predict what will happen next. Why does this pattern keep happening ? Because we try to model things on what could be nice than what is there IRL. As the real world is not nice these mathematics accidentally shoot their feet. When something does not work you change the model, but when your foundation is based on things like “purity”, vanity and psuedo-science you blame everything else but the model. Hint: Linked List supremacy is a silly idea. Water has the shape of the container it takes.
I think pure mathematics is ripe for a Sokal scandal. We need A.I to design a paper that is a 100 pages long, unreadable by humans because it tests human memory limitations and unprovable by computers because it has to learn something new.
I used to think that majority is wrong and the minority is right. Now days I think majority is right sometimes and the minority is right sometimes. When the majority is wrong and evil, you will know it by their fascistic tendencies. When the minority is wrong and evil you will know it by there colonising and holier-than-thou attitude. Forth and Erlang represent authentic minorities in this sense. In programming forums I have never seen C++ programmers exhibit delusions for just using objects. You will notice these authoritarian tendencies when these thin-skinned fanatics bitch and moan about minor issues. The greatest trick of colonial minorities is how they infect the education system because they sure as hell can’t compete in the market place. Haskell fanboys can’t write an operating system but they will sure teach you about how computers really work and tell you that pointers and assignments are evil.
We need to reject the need for category theory in computer science as cancer. I think there is scope for new type system based on fuzzy theory, quantum theory and multi valued logic. I think programming notation can liberate modern mathematics. But we are at risk of losing all the progress made by programmers if we let the intellectual colonists tell us that BASIC is worse and we should all be “ashamed” and that we all should learn Java, Haskell and Lisp because it represents real programming because so many operating systems, applications and games are written in these sublime languages. More than any, we risk making programming a joyless bore and ritual dribble from the lips of high school teachers and superstition for dogma.