## Mathematics is the part of physics where experiments are cheap

• 169
This is a citation from "On teaching mathematics", by V.I. Arnold
(https://dsweb.siam.org/The-Magazine/All-Issues/vi-arnold-on-teaching-mathematics)

In my opinion what he says makes perfect sense.
However, I believe this is the opposite of the mainstream point of view on mathematics, both for mathematicians and philosophers.

So, I would like to know if somebody has some convincing arguments against this point of view.
• 249
Mathematics is the part of physics where experiments are cheap, and brain power is at its most expensive.

Most maths that drain brain-cells can cost up to \$302,595 per hour. This figure involves not only the pay to the actuary / mathematician, but the cost of training and the cost to society as well.

This figure goes up and down, depending on the current demand on math knowledge, and computer program sophistication.
• 14
Mathematics is the natural language physics is written in. The validity of a physical theory is an experimental matter - nothing to do with the math.

Here is an interesting example:
https://arxiv.org/abs/1507.06393

It's mostly just math - but there is that word - almost - in there. That almost is the existence of magnetic mono-poles which is an experimental matter - and far from easy to experimentally look for. None have yet been found - but may in the future in which case Maxwell's equations are not quite correct - however its easily accommodated in the math of the linked paper.

Another example is Lovelock's Theorem that shows in dimension 4, the Einstein equations are the unique second-order field equations generated by an action. i.e., you can derive Einstein's equations by requiring that they are generated by an action and that they are of second order in the derivatives of the metric. Normally actions contain only first order derivatives but its impossible to construct a first order one in GR, so you go to second order. The interesting thing is it turns out when you calculate the GR equations the second order derivatives in the action do not matter, and you get normal field equations that second order actions would not usually give. Now the question is why do we have actions (I will not go into why you would usually only want first order derivatives in the action)? That requires QM to explain (it follows from Feynman's path integral approach). Without the supporting experiments who would come up with QM?

The relation of math and physics is in fact quite subtle.

I do however find the ending of the link in the first post amazing - its true - but virtually never pointed out except by those like me that have read it:
'A teacher of mathematics who has not got to grips with at least some of the volumes of the course by Landau and Lifshitz will then become a relic like the person nowadays who does not know the difference between an open and a closed set.'

The first book in the series, mechanics, is simply beauty beyond compare as reviews on Amazon attest to:
https://www.amazon.com/Mechanics-Course-Theoretical-Physics-Landau/dp/0750628960

BTW it explains, amongst other things why actions only (usually) contain first derivatives.

Could this be taught at HS? Well its deep and requires multi-variable calculus but IMHO it can - generally where I am in Australia calculus is taught later than it should (for good students its taught by some schools in grade 10 - so it is possible for them - normally one waits until grade 11 and 12). Then books that authors think well prepared HS students can handle like Morin's is possible:
https://www.amazon.com/Introduction-Classical-Mechanics-Problems-Solutions/dp/0521876222

Thanks
Bill
• 56
I don't see to much validity in his general point at all.
To preface my point: I am not a mathematician nor a physicist. Therefore I don't doubt the validity of his mathematical statements. However they seem to be rather disconnected from the main thrust of his argument.

Maybe it makes most sense to contrast his view with my understanding of math. I understand math to be a semi-language (please ignore the semi it doesn't add to the point here).
I assume this to be the case because in general natural sciences transform their observations/measurements of reality into math to be in a precise framework where predictions and models are precise and with which they are testable.
It certainly seems true that math comes to play in basically any natural science not only physics. If one does not hold the unproductive view that all science is physics which seems to be a error in categorizing and therefore setting the border of physics, it seems obvious that his statement is just painstakingly oversimplifying the question.
I want to point out that one can still endorse sub-points he raises that are due to having problems with translating physical phenomena into math or if one uses bad conceptual frameworks for some real word applications.
There is also the problem of over evaluating math, assuming it to be something it is not making it infringe on the domain of physics.

Math gains its precision from being a purely axiomatic, and therefore deductive system, in which there are valid transformations and invalid transformations. The degree of how useful specific transformations and certain areas of math are depends on the framework.
There simply aren't any experiments in an axiomatic system. Stating this painfully reveals a misunderstanding of what axiomatic systems are. This gets way more obvious if you look at logic that is an axiomatic system that is at the core of math. So I want to illustrate this point with logic rather then math because it makes the point clearer.
Take:
Socrates is a Dog
All Dogs are Mortal
Socrates is Mortal
Logic and similarly math investigates if the transformations and therefore the conclusions are valid. It doesn't say anything if it is sound. After-all Socrates could not be a dog.
Meaning logic says giving the premises are true(which we don't know for certain/could also be framed wrong) the conclusion is true. There is no statement about the concepts "Socrates,Mortal,Dog" and limits itself to interpreting "all,is" assuming that all is used correctly as all and not as most or something else.

I personally hate it when philosophers use math examples of the type "1+1=2" but here it seems to fit.
He is complaining about students transforming 2+3 into 3+2, however this is not false and the transformation signified by =(valid transformation from one side to another) holds. The kids could instead also have said 2+3=1+1+1+1+1. I make this point because it clearly shows that he is complaining about the usefulness of using this transformation. However this could be the best transformation if you want to prepare the numbers to get fed into a computing machine.

In conclusion I don't think his argument as he put it (not as I interpreted it with usefulness) is clearly wrong on so many levels. Math is not limited to physics, not everything is physics (a conclusion one would have to draw from his statement), axiomatic systems are not internally experimental in any sense(only maybe in the choosing of the axioms), he doesn't argue his point properly which also would have to be extendable to logic(and instead babbles long about non trivial concepts in math that are used in specific contexts to intimidate the inexpierenced reader and to illegitimately gain authority by over complicating things)(Note I so harshly point this out because if he takes out the right to assert inferiority complexes and other insults and uses this kind of argument I will to)

Last but not least I want to stress that his core point is correct but not because of his faulty arguments and his missuses of concepts. Take F.e. a look at his first two sentences based on my argumentation that actually addresses the issue
"Mathematics is a part of physics.(Wrong/misuse of concepts) Physics is an experimental science(what science isn't? biology?), a part of natural science. Mathematics is the part(same mistake as before) of physics where experiments(there are no experiments in axiomatic systems) are cheap.(why should cost play any role at all in his argument?)"

However it is obviously very problematic disconnecting a language from its field of application. Just like it seems to be problematic that the language experts have no proper education in the field of application leading to not useful transformations. This point seems more obvious for natural languages then math (due to it being more abstract) however it still holds.
Btw. Him mainly complaining about the french education system and extending it to an "universal hatred of mathematicians" or interpreting the pupil that transformed 2+3 into 3+2 as "not know[ing] what the sum was equal to" contains such obvious misconceptions or at best questionable rhetorical tactics that they alone should lead one to question the article.

Obviously I'm open to be shown wrong or to someone pointing out where my argument is not convincing.
• 441
A theory of physics is fundamentally a method that allows to make predictions from observations. We don't have to see mathematics as an underlying necessity to formulate these methods, we can simply see it as a tool, a tool of thought (I think it should be possible to construct accurate methods that do not rely on mathematics as we know it, but say on pictures or on videos).

A given method can be formulated more simply in some mathematical formalism than in some other, that is using some tool rather than some other, so in that sense experimenting on mathematics can sometimes make it easier to carry out the method that turns observations into predictions. But then I wouldn't ascribe to mathematics some more profound significance than it being a tool of thought, I know some see mathematics as the language of the universe but I don't agree with that view.

Occasionally experiments in mathematics might indirectly lead to new breakthroughs in physics, for instance if the simplicity of a formalism allows to look at a problem in a way that was not possible using a more complicated formalism. But I think that also brings its fair share of potential problems, because when we focus too much on the mathematical formalism we become removed from the physical experiments, and then we forget what made the formalism useful in the first place and we start reifying it as having primacy over experiments.

For instance enormous thought resources have been spent in trying to unify general relativity with quantum field theories, with a focus on mathematics, and not so much on attempting to account for experiments in a different way than through these formalisms. Physicists mostly build on top of the mathematical formalisms of previous theories, instead of starting afresh and attempting to explain experiments without the mathematical baggage of the previous theories.
• 7.7k
there's a joke I learned on philosophy forums. It's about an interview between a University Vice-Chancellor and the Budget Committee about funding. The VC is going through budget requests; Physics has requested squillions of dollars for some apparatus regarded as vital to the Uni's future.

VC: Physics Department is driving me round the bend.
Committee: What's the problem?
VC: They want so much money! Like, tens of millions! Apparently they need some kind of splitter to keep up with Harvard.
Committee: Do we have the funds?
VC: Well, only if Physics gets everything, and everyone else gets nothing. Why can't they be more like the maths department? They only want papers, pencils, and wastepaper baskets.

Or the philosophy department? They don't even want the wastepaper baskets!
• 82
This is a citation from "On teaching mathematics", by V.I. Arnold

V.I. Arnold is a constructivist heretic.

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry.

Mathematics supplies its consistency-maintaining bureaucracy of formalisms to more than one other field. It is not just the private prostitute or concubine of physics. What about people in engineering, business, computer science, and industry?

Mathematics is a part of physics.

You see !? The article already starts with designating mathematics as the private whore of physics.

Mentally challenged zealots of “abstract mathematics” removed all the geometry

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.

Secondly, Carl Friedrich Gauss algebraically described the limitations of geometrically constructing numbers with unmarked straightedge and compass:

Since the field of constructible points is closed under square roots, it contains all points that can be obtained by a finite sequence of quadratic extensions of the field of complex numbers with rational coefficients.

In order to compute numbers that lie outside these quadratic field-extension towers we must use algebra. Geometry cannot handle such numbers, such as cube roots, or even roots of arbitrary power, and so on.

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.

The scheme of construction of a mathematical theory is exactly the same as that in any other natural science.

His constructivist heresies are nauseating!

Then we try and find the limits of application of our observations by seeking counter-examples ...

That is physics. That is not math!

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”

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).

The mathematical technique of modelling consists in ignoring this trouble and speaking about your deductive model as if it coincided with reality.

Mathematical systems (not "models") are not even meant to coincide with reality. Mathematics never says anything at all about the real world.

His views are heretical!

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.

And he is a criminal constructivist!

Axioms are unmotivated definitions because any requirement to motivate them would lead to infinite regress. As Aristotle said: "If nothing is assumed, then nothing can be concluded."

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.

By shedding Euclid's Elements, mathematics has finally become pure reason, which is language (symbol manipulation) only. We do not want so-called usefulness. We want purity, because ultimately, it is purity that is math's usefulness.

... university mathematics courses (from which in France, by the way, all geometry has been banished in recent decades).

Yes, the impurities of visual puzzling had to be stopped. Good riddance. Math had to become pure reason, i.e. language only.

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.

Math is staunchly axiomatic. That is non-negotiable. Seriously, get over it!

Someone had better tell this author to come to grips with the fundamental epistemic method of math; and if he does not like it, then he should stick to physics instead.
• 14
Some interesting points here. What I would like to do is have a look at a concrete example. Consider 1+2+3+4......... It's obvious to most its infinity. But consider the so called Zeta function:
https://simple.wikipedia.org/wiki/Riemann_zeta_function

Let s = -1 and you get 1+2+3+4......... . and it is known that at -1 the Zeta function is -1/12 or 1+2+3+4......... = -1/12. Even stranger is calculations such as the Casmir Force uses 1+2+3+4......... = -1/12, and it is in accord with experiment. A very strange phenomena - or is it? I have my view but would be interested in what others think.

Thanks
Bill
• 169
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.

Yes, I know Hilbert's program, from the beginning on 20th century (https://plato.stanford.edu/entries/hilbert-program/):
all mathematics should be reformulated as pure formal logic: syntactical operations on strings of symbols, following a well defined set of rules. The meaning of symbols is completely irrelevant in proofs. If a proof depends in some way on the meaning that you give to the symbols, it means that you are making hidden assumptions that should instead be expressed in a purely syntactic form.

And today we can be sure that this works: there are several extensive libraries of formalized proofs making use of computer-based formal logic systems, and a simple personal computer can verify the correctness of thousands of theorems in a few minutes. I guess this is Kant's "pure reason" in it's purest form.

But, as you probably know, there is a part of Hilbert's program that instead didn't work, despite more than 50 years of efforts to realize it. The idea is that if logic becomes a purely formal game, the search of a proof can be reduced to a mechanical computation: just search for all possible formal proofs and verify their correctness one by one, until you find one that is correct.
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.

I believe that this fact is becoming more and more clear with today's rapid development of artificial intelligence. And this is the same problem that V.I. Arnold is referring to in this article, when he speaks about teaching mathematics: you cannot teach mathematics as a purely symbolic game, because in this way it has no meaning at all.

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.

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.

You are right: you can use algebra to generalize geometry. But you need geometry to have something to be generalized: the axioms and definitions of algebraic geometry are carefully chosen to correspond to Euclidean geometry as a the particular case.

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).

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.

We do not want so-called usefulness. We want purity, because ultimately, it is purity that is math's usefulness.

Yes, we don't want usefullness, but we want beauty, simplicity, symmetry, elegance. Try to take any of the most elegant parts of mathematics (say, for example, the theory of holomorphic functions - or complex functions' calculus), write it using ZFC first-order logic, changing all the names into random strings and present it to a mathematician without explaining what is it. Well, maybe I am wrong, but I think that nobody would say that this is beautiful mathematics.

I'll stop here for now (I don't want to make this post too long), but I hope I made clear what I mean...
• 82
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.

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.

For example, in physics, the semantics concern a deeper understanding of the real, physical world. Therefore, physics is supposed to be semantically oriented and semantics-heavy. If people are interested in understanding the real world, they'd better do something like physics, and not pure mathematics.

you cannot teach mathematics as a purely symbolic game, because in this way it has no meaning at all.

Agreed. There is an important difference between teaching versus proving mathematics. Even though semantics are banned from being used in proofs, they are actually ok elsewhere. I only objected to V.I. Arnold suggesting that the epistemic justification method in math should be empirical.

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.

There may be a danger in encouraging people to look for semantics/meaning in mathematics itself instead of picking a physics theory such as Maxwell's equations.

(Pure) mathematics is for people who appreciate the beauty of formalisms and surprising structures that emerge out of the design of abstract, Platonic worlds. It really is a goal in itself, and it is supposed to be done with total disregard for possible applications.

It is a particular type of sensitivity and talent that some people have and others not really.

Why would everybody need to study subjects that they may not even like?

There are so many other fields of endeavour where people can find satisfaction in pursuing understanding. I think that it is wrong to force everybody to learn more than just basic arithmetic, just like it would be wrong to force everybody to every day spend hours in learning how to play the violin.

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.

Well, we actually do it all the time.

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.

I always thought that two symbols was the minimum, but now we know that just one is enough.

By the way, combinator calculus is undoubtedly "useless" in a sense of not having any applications. To tell you the truth, I actually prefer systems that do not readily have applications waiting for them. If these applications are eventually found anyway, they tend to be extremely powerful. Low-hanging fruit, on the other hand, tends to be rather worthless on the long run.

By the way, a "combinator" has no meaning. It is just a symbol in the system. I like it that way, because that kind of symbols should have no meaning. Combinator calculi are indeed a collection of meaningless theories. I like it that way.

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. Almost every non-trivial piece of software now uses it. The following is a popular portal for regular expressions, and this is an online testing tool for RE.

Regular expressions (RE) have no "meaning". They are just syntax patterns. Still, they often trivially solve otherwise really big problems in software. If RE actually had a meaning, they would not even be useful ...
• 169
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.

I totally agree that the use of logic in mathematical proofs, and the successive refinement of logic to formal logic at the beginning of 20th century, has been one of the one most important developments in the history of science. And it's this development that has given mathematics the freedom to create "new worlds", as Grothendieck used to say (http://www.landsburg.com/grothendieck/mclarty1.pdf).

But from another point of view, the evolution from ancient Greek's mathematics regarding Euclidean axioms of geometry as "a priori" to modern mathematics regarding symbolic logic as "a priori" is due to the recognition (due mainly to Riemann) that geometry should be really regarded as a branch of physics, and the facts that space is 3-dimensional and Euclid's 5th postulate is true should be regarded as experimental facts.

Well, what if the same was true even for symbolic logic?
If you think about it, what makes formal logic proofs absolutely certain, as opposite of geometric constructions, is the fact that they can be interpreted as algorithms or, equivalently, as purely topological geometric constructions (graphs built following a precise set of rules). And we are sure that every time that we execute the same algorithm with the same input, we get the same output (or, equivalently, that every time we follow the same path on a graph we always get to the same point).
But what if we leaved in a world where deterministic laws of physics do not exist? As we know from quantum mechanics (if quantum mechanics is correct), all laws of physic are in reality probabilistic: in principle you can never be sure with absolute certainty that the same experiment will give you always the same result. So, the reason why formal logic can be used to prove theorems with absolute certainty depends ultimately on physics: this is because at the macroscopic level the laws of physic are deterministic with an extraordinary high degree of approximation.

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.

The Iota combinator is interesting because it's the simplest possible calculus that is equivalent to lambda calculus. And lambda calculus is interesting because it's equivalent to turing machines, and both are representations of the fundamental concept of a total recursive function.

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.

Kleene's closure is interesting because is a model of the algebraic structure called monoid, and a monoid is interesting because it's the generalization of a group. And a group.. you surely know.
• 169
Even stranger is calculations such as the Casmir Force uses 1+2+3+4......... = -1/12, and it is in accord with experiment.

Do you have a reference to a document where this result (1+2+3+4......... = -1/12) is actually used to calculate the the Casmir Force ?

A very strange phenomena - or is it? I have my view but would be interested in what others think.

The normal explanation, as you probably know, is that the the Zeta function coincides with the infinite series only where the series is convergent ( Re(s) > 1 ), but for s = -1 is defined as an integral, and the integral is convergent (https://en.wikipedia.org/wiki/Riemann_zeta_function).

The interesting question is: is there a way to define complex numbers so that the infinite series coincides with the integral on all complex plane? I don't know, but I even don't know any proof that this is not possible. Surely, to make this work complex numbers cannot be defined as an extension of the standard real numbers.
• 169

when a theorem is discovered, the theorem is conjectured to be true on the base of some physical intuition, or generalization of previous results, and then axioms are found to make the proof work.

when the same theorem it taught to the students, the axioms are presented (and you usually do not see why those particular axioms should be considered instead of others), and then the theorem is proved by means of logical deductions, sometimes without even mentioning the physical intuition and generalizations that was the base of it's discovery.

It is true that mathematics can be treated as the application of formal logic (the language) to a set of axioms, but in this way you lose the most important part of it: it's intuitive meaning.
• 249
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.

I don't find any reason to accept that pure reason must be ONLY language-based.

After all, language is a symbolic medium, and it could not exist without a basis of visual and other sensory input. Without senses a person could not learn to speak.

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.
• 82
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.

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.
• 249
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

This is a very neat description of your admiration for Kant. However, your post did not take anything away from my criticism of his finding that pure reason must be lingual only.

You are a puppet of the traditional dogmatic Kant-cult. You can't come up with a single argument against a simple argument that single-handedly destroys his thesis. Yet you worship him. Typical cultism. No reasonable argument can daunt your devotion to him.

Why?
• 14
Ok - first an example of its use in calculating the Casimir Force:
https://pdfs.semanticscholar.org/90a0/5e9d920f271f20704406828d94d0890c6608.pdf

See equation 2.31.

The above uses a naive summation, a deeper justification based on renormalization can be found here:
https://www.iopb.res.in/~sjp/sjp_past_issues/sjp_aj17/new/Torode_Casimir.pdf

The idea of re-normalization is this. You do calculations, but end up with infinity. To get around this one writes it in a form with a new parameter called a cutoff that gives a finite answer - a wrong one of course because you have a cutoff in there - but an actual answer depending on the cutoff. Give this cutoff dependent answer the name re-normalized value - a fancy name for a simple concept. Rewrite the equation in a different form by some algebra so it contains the re-normalised value but no explicit cutoff (the cutoff is now contained in the re-normalized value). That way you can do calculations that give finite answers. Now take the limit as the cutoff goes to the correct value. But the cutoff is now contained in the re-normalized value and we know what that is in the limit because we can measure it. Tricky - here is a paper on it:
https://arxiv.org/abs/hep-th/0212049

Now in the Casimir force you do a slightly different variation of the trick. You write the infinite sum in terms of the Zeta Function with s as the parameter in the Zeta Function. Its only defined for s > 1 but we want it for s = -1. Its the same trick as above but here its much easier. You simply let s approach -1 which can be done by a powerful technique called analytic continuation that can be done with complex numbers - but not real numbers unless considered as part of the complex plane.

And therein lies what is going on. If I write 1+2+3+4......... I make an assumption not implied by the the equation - that the numbers are real. However they can equally be taken as complex numbers. In the complex plane the powerful method of analytic continuation is available to use and it can be summed. Math is not just about logical constructs - its more than that - its about concepts. In order to do math we must choose the right concepts and interpret your equations using those concepts.

It took me a while in my mathematical studies to nut that one out. My professors used to get annoyed at me because I asked questions about logical issues. But gradually I realised while math requires rigorous logical proof it is not done in a vacuum - but has a context and that context has a big impact.

Thanks
Bill
• 683
So, I would like to know if somebody has some convincing arguments against this point of view.

Great essay and I've read some of his other stuff too. It's not the kind of thing one must vociferously disagree with. It's in the category of "Interesting even if wrong." Even if I believed that math is pure and must remain unsullied by practical concerns (an extreme version of formalism I suppose) I wouldn't disagree with Arnold about anything.
• 82
However, your post did not take anything away from my criticism of his finding that pure reason must be lingual only.

Statements of reason, pure or not, will have to be expressed in language. Otherwise, they cannot be communicated.

The term "purity" in Critique of Pure Reason means: no sensory input. Accepting visual, auditive, tactile, smell, or any other such signal in the process of producing a theorem, is not allowed.

Kant already clarified that using a straightedge and compass, as advocated in Euclid's Elements, to draw geometric figures, and then solve visual puzzles in them to produce a theorem, simply amounts to accepting sensory input, and is therefore not to be considered pure reason.

Nowadays, we have an alternative approach to geometry that is fully algebraic, i.e. language only, and that therefore successfully addresses Kant's objections.

Is there anything else that you can use, besides language, that will properly stay clear of accepting real-world input? In my impression, there isn't.
• 249
Okay, so if Kant said pure shit is something that comes out of everyone's mind who writes it down, then we'd be exchanging shit?

So Kant all of a sudden took ownership of reason and of pure reason, and whatever definition he gives it, stands?

You a sheep or a human?

HIS nomenclature applies to HIS writings when he makes references to HIS definitions, but Kant ought not to have hijacked the language. If he says "pure reason" he means written communication. But it does not, ought not to transfer to common language, to the language other people have long before him agreed to use in the way they do.

Statements of reason, pure or not, will have to be expressed in language.

This, my dear Alcontali, is false. Reason can be communicated using many, many different forms of communication, other than lingual.

For your information, there are IQ tests even that separate different levels of reasoning ability with nothing but diagrams. Kant would have committed suicide if he heard of this. 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.

Please think things through before committing to their alleged truth.
• 169
Hi Bill,
I read the paper on Casimir Force. Well, there are several passages that I don't really understand, like for example the derivation of (2.36) from (2.34) (let alone the funny use of zero exponent). But I don't see any use of the Riemann zeta function for s = -1. He completely omits to write limits on complex variables, but I think that if you put the limits back where they should go, he didn't use divergent series treating them as convergent.

Anyway, I saw the use of divergent series several times ( Euler and first of all Ramanujan ), and it's really very strange that using algebraic manipulations on formulas that should be without sense when interpreted as sums of natural numbers they can obtain anyway the correct result.
It seems that there really should be some interpretation on these formulas that has to make sense and is not natural numbers. But I have no idea what that interpretation could it be... do you?
• 82
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.

Numbers are still language. Diagrams are not.

Bourbaki sought to ensure the purity of mathematics by axiomatizing and algebraizing. Bourbaki does not make explicit reference to Kant, but reaches the same conclusions as Kant about the use of visual illustrations:

It is fairly clear that the Bourbaki point of view, while encyclopedic, was never intended as neutral. Quite the opposite: it was more a question of trying to make a consistent whole out of some enthusiasms, for example for Hilbert's legacy, with emphasis on formalism and axiomatics.

Furthermore, Bourbaki makes only limited use of pictures in their presentation. Pierre Cartier is quoted as later saying: "The Bourbaki were Puritans, and Puritans are strongly opposed to pictorial representations of truths of their faith." In general, Bourbaki has been criticized for reducing geometry as a whole to abstract algebra and soft analysis.

While several of Bourbaki's books have become standard references in their fields, some have felt that the austere presentation makes them unsuitable as textbooks.

Of course, even though pictorial representations may be legitimate to aid understanding, it is necessary to prevent students to involve them in formal proof; which must be text only. Therefore, I do support Bourbaki's iconoclasm.
• 14
He has really goofed it up - look at equation 2.26 instead. If it was to the power 0 then you would have 1+1+1+1..... which in fact is -1/2 and his answer is wrong. I would need to do the math correctly from 2.26 to see what he is getting at. The point is 2.26 contains the Hurwitz Zeta Function with s = -1:
http://mathworld.wolfram.com/HurwitzZetaFunction.html

As you can see from the above the equation he wrote later connecting it to the zeta function is correct and that is how the Zeta Function at -1 or 1+2+3+4..... enters into it. I just hate it when a supposedly peer reviewed paper has errors like that - but it happens a lot more frequently than people think.

My view is as I said. When writing something like 1+2+3+4...... because you are using real numbers you think you are confined to the real number system. But there is nothing stopping them being from the complex number system in which case you have analytic continuation that allows the summation to be defined. A similar trick is used in a number of summation methods in order to interpret whats going on (the following also contains a correct derivation of the Casmir force using the Zeta Function - I checked it this time):
https://arxiv.org/pdf/1703.05164.pdf

Take Borel Summation as an example. Deriving it is simple as well as seeing its a natural extension of normal summation. Its always true if you can reverse the sum and integral. For normally convergent sums there is a point where for all practical purposes the rest of the terms are zero so you can to any accuracy replace the sum by a finite one hence the order can be reversed. If not normally summable you have a method for summing series you normally can't sum. Whats really going on? Although it only uses real numbers by going to the complex plane you can see its a hidden method of analytic continuation which, as explained in the link is mainly what divergent series summation is.

Math is funny - looking at exactly the same problem in a different context, often one that is more general, allows a deeper understanding.

BTW the above paper is by the same lecturer that did a series of lectures containing a lot of the above plus all sorts of puzzling mathematical physics stuff worth philosophical consideration. I will admit I am not enough of a philosopher to nut it out.:

Thanks
Bill
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