This is a citation from "On teaching mathematics", by V.I. Arnold — Mephist
Mathematics is a part of physics.
Mentally challenged zealots of “abstract mathematics” removed all the geometry
The scheme of construction of a mathematical theory is exactly the same as that in any other natural science.
Then we try and find the limits of application of our observations by seeking counter-examples ...
certain facts which are only known with a certain degree of probability or with a certain degree of accuracy, are considered to be “absolutely” correct and are accepted as “axioms”
The mathematical technique of modelling consists in ignoring this trouble and speaking about your deductive model as if it coincided with reality.
Attempts to create “pure” deductive-axiomatic mathematics have led to the rejection of the scheme used in physics (observation, model, investigation of the model, conclusions, testing by observations) and its replacement by the scheme definition, theorem, proof. It is impossible to understand an unmotivated definition but this does not stop the criminal algebraist-axiomatizers.
Any attempt to do without this interference by physics and reality with mathematics is sectarian and isolationist, and destroys the image of mathematics as a useful human activity in the eyes of all sensible people.
... university mathematics courses (from which in France, by the way, all geometry has been banished in recent decades).
If mathematicians do not come to their senses, then the consumers, who continue to need mathematical theory that is modern in the best sense of the word and who preserve the immunity of any sensible person to useless axiomatic chatter, will in the end turn down the services of the undereducated scholastics in both the schools and the universities.
Firstly, Immanuel Kant pointed out in his Critique of Pure Reason that the practice of solving visual puzzles, as in Euclid's Elements, could not possible be considered pure reason, because it rests on fiddling with visual input, while pure reason must be language only, entirely devoid of sensory input. That is one reason why an algebra-only, pure-reason approach to mathematics is much preferable to geometric fiddling with visual puzzles. — alcontali
Geometrically constructible numbers are just a very, very small subset of all computable numbers. Therefore, these Euclidean methods hold us back. We had to drop them, in order to be able to progress. — alcontali
Axioms are not "correct". Axioms are just arbitrary starting points for the construction of an abstract, Platonic world.. Axioms have nothing to do with the real world (just like everything else in math). — alcontali
We do not want so-called usefulness. We want purity, because ultimately, it is purity that is math's usefulness. — alcontali
This idea doesn't work for one fundamental reason: once you have depleted a mathematical proposition of any meaning, you have no clue why that theorem should be true, or should be distinguished from the infinite sea of combinations of symbols that can be interpreted as theorems. — Mephist
you cannot teach mathematics as a purely symbolic game, because in this way it has no meaning at all. — Mephist
In other words: the meaning of a theory is not contained it it's purely symbolic representation, but in it's correspondence with the way the physical world works. In this sense, algebraic geometry (for example) is not substantially different from Maxwell's equations. — Mephist
It is true that from any set of axioms you can build a theory and derive the relative theorems, but I guess that nobody would be interested at all in axioms that do not correspond to any generalization whatsoever model that corresponds to ideas taken from the physical world. If you choose the 'wrong' axioms, you obtain a meaningless theory. — Mephist
There is also a merely mechanical reason why it does not work: Gödel incompleteness theorems. I am actually not against the use of meaning, i.e. informal semantics, in mathematics. I am only against the use of semantics as proof; which should be syntactic only.
Actual meaning will be plugged in by the discipline that applies the mathematics. — alcontali
For example, Chris Barker's Iota combinator calculus is a Turing-complete system with just one combinator, i.e. Iota. People already wondered if the SK combinator calculus could be simplified from two symbols to one. So, Chris Barker positively answered that question. — alcontali
Kleene's closure is an example of a theory that was utterly useless for a very long time, but surprisingly beautiful, and even intriguing. In the meanwhile, it has turned into a rage, some kind of hype. — alcontali
Even stranger is calculations such as the Casmir Force uses 1+2+3+4......... = -1/12, and it is in accord with experiment. — Bill Hobba
A very strange phenomena - or is it? I have my view but would be interested in what others think. — Bill Hobba
Firstly, Immanuel Kant pointed out in his Critique of Pure Reason that the practice of solving visual puzzles, as in Euclid's Elements, could not possible be considered pure reason, because it rests on fiddling with visual input, while pure reason must be language only, entirely devoid of sensory input. — alcontali
Kant was a ninnie. I've been harping that forever, but nobody pays me any attention. Instead, they turn to me and say, "how can you say that? BLASPHERMER!" Whereas all you have to do is read what Kant wrote, think about it for five minutes and you realize that the bloke was full of false views. — god must be atheist
I think that Kant is the greatest epistemologist ever to have set foot on this earth. I also consider him to be the first epistemologist to have made real progress after Plato and Aristotle. As far as I am concerned, after him, there are only Karl Popper and Edmund Gettier to have contributed meaningfully. Epistemology is a field with very few names to mention. There have been lots of philosophers but only a handful of them have managed to do something meaningful in epistemology.
5 hours ago — alcontali
So, I would like to know if somebody has some convincing arguments against this point of view. — Mephist
However, your post did not take anything away from my criticism of his finding that pure reason must be lingual only. — god must be atheist
Statements of reason, pure or not, will have to be expressed in language. — alcontali
There are text books with numbers, that would never pass as reasoned texts if you took the numbers out of it. Same with diagrams in textbooks. — god must be atheist
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