Dear Mr. Reuben Hersh,

There is a question that has been troubling me for a long time, and I would be very grateful if you could shed some light on this.

A Nobel Physics Prize judge (H. Nordenson) says "it is a normal experience that when somebody attempts to criticise or even analyse the fundamental principles and/or definitions of the Theory of Relativity, he is inevitably told that Einstein’s theory is of a purely mathematical nature and that anyone who is not a specialist in mathematics is excluded from the possibility of grasping its contents, that he is unduly prejudiced and fettered by old traditions and ideas and that he is innately alien to these revolutionary lines of thought and therefore more or less incompetent to deal with them."

Well, my question, followed by several related to it, is:

How come that a theory which is telling us about the deep nature of the world we live in cannot be expressed nor grasped through verbal thought and language, which is richer, more nuanced and more versatile than mathematics, and from which ultimately the latter sprung up? Aren’t physicists in general relying too heavily in mathematics and too contaminated by strong "mathematical Platonism" inasmuch as they seem to be so convinced that Galileo was right when he stated that "the book of nature is written in mathematical form"? Isn’t mathematics at bottom just a system of formal symbolic logic (specialised to deal with quantities) and accordingly just an expression of human reason, and shouldn’t be any mathematical equation -- at least its meaning -- expressible in ordinary language? Is it possible that anything excepting purely mathematical problems could be expressible only through mathematical language? How could we humans ever learn, teach or develop mathematics if it was not graspable by human reason, if it didn’t make any sense, if the meaning of a number or a formula was not verbally expressible?

As far as I know, mathematics is nothing but a linguistic-symbolic tool developed to operate with quantities -- countable and measurable quantities. I dare to say (please correct me if I’m wrong) that taxonomically, despite maybe not historically, mathematics could be regarded as a branch of formal logic, which in turn is nothing but a useful systematisation of the natural forms of human rational thought. I can’t see any reason to suppose that even the most arcane mathematical expression, equation or symbol, as long as it makes any sense, could be not explainable in plain verbal language, since either you can say that mathematics arose (or could have arisen) from verbal logic or you can say, at the least, that both languages share the same syntactical and logical structures deeply ingrained in the human brain.

I’m looking forward to hearing you comments about that.

Best wishes,

Fabio Fernandez

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Dear Fabio Fernandez,

I’m afraid that what Prof. Nordenson says is a "normal experience" is news to me. I can’t comment on his strange unsupported accusation.

Mathematics is not just part of logic, and it is not just a language. I have written a book called "What is Mathematics, Really?" published by Oxford. There I gave my best explanation of what mathematics really is.

In principle, anything in mathematics can be restated without mathematical terms and symbols. We use mathematical terms to make understanding easier, not harder. For instance, try doing a tax return or any other complicated commercial account without using the mathematical terms "plus," "minus," "equals," "one," "two," "three," and so on.

Or do elementary geometry without using mathematical terms like "point," "line," "circle," "similar," "congruent," "parallel", "intersect", and so on.

It would be possible to explain electricity and magnetism without the words "charge," "voltage," "resistance," "pole," "positive," "negative," "attract," "repel", and so on, but it would not make it easier.

Perhaps you object that these words are ordinary language words, not technical words. But their use in mathematics and physics requires giving them precise technical meanings not at all the same as in ordinary language.

To go beyond arithmetic and elementary plane geometry requires introducing new concepts. We may give them common English names like "group," "ring," "field," "imaginary," "complex", "projective," "category," but now the common meanings do not help at all to understand the mathematical concepts that bear these names. Perhaps it is better for the student when we use new terms like "isomorphic" or "Hilbert space" or "Lie group." In any case, there is no substitute for the time and effort to actually study and learn the content of any subject one wishes to master, let alone to contest or criticize.

Sincerely,

Reuben Hersh

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Dear Mr. Hersh,

Please let me insist a little bit more, because I think maybe I didn’t manage to be clear enough about my true worries in my previous e-mail (to which your answer was very fitting and fulfilling) and also because the issue of the philosophical and logical foundations of Relativity (and perhaps also the issue of the extent of the role that mathematics shall play in physics) is a very important one that has been debated in the background for decades and tends to become increasingly hot in the next years, as the physical community starts to feel desperate in face of the cul-de-sac arrived at in quantum gravity research (in the words of mathematical physicist John Baez: "About the only thing that everyone working on quantum gravity agrees upon is that general relativity is just an approximation. It must be, because it doesn’t take quantum mechanics into account, and the world is quantum-mechanical. So the big question is: how radically must we break from the picture of spacetime provided by general relativity? It makes sense to try the most conservative things first, then if those don’t work, more radical things, and so on. People have been working on this for about 50 or 60 years, so by now they are getting desperate and trying some fairly radical things. In the conferences on quantum gravity that I went to earlier this spring, I noticed a surprising unanimity of opinion about one thing. People from string theory, loop quantum gravity, noncommutative geometry and so on disagreed about almost everything, but they almost all seemed to agree that we need to move away from the picture of spacetime as a manifold.") Anyway, I used Prof. Nordenson’s argument and the problem of Einstein’s Relativity just as an illustration, but I could have as well picked up Hawking’s assertions about "the beginning of time" in his Big Bang model or any other arcane and quasi-mystical contemporary physical theory based on metaphysical assumptions whose veracity can allegedly be proved only through mathematical means. My true question, the one I was indeed trying to pose and probably failed to formulate clearly, is:

To what extent, if any, is mathematics able to prove the veracity of metaphysical propositions? (Like, say, the existence of time as a real physical force or entity or the existence of God.)

Because the fundamental underlying starting assumption in Einstein’s Special Relativity -- the assumption (later reinforced by Minkowsky’s spacetime geometry) that time is a real physical force or entity, since it must be so in order to be susceptible of slowing down, speeding up or stopping -- is indeed a metaphysical assumption (or physical, if you like), which if proved untenable could compromise or even collapse the entire Relativity's logical and "aesthetical" edifice (although not necessarily its "workability").

The history of Relativity’s genesis can be roughly summarised as follows:

A) the Michelson-Morley experiment tells Einstein that the speed of light is the same in any inertial reference frame.

B) Einstein infers that the only possible explanation to this astonishingly surprising and disconcerting aberration of nature is time dilation.

C) for time to be capable of dilation it must in the first place exist as a real physical entity or force.

Well, here we have the crucial metaphysical assumption, which instead of a conclusion arrived at through mathematical means is a fundamental starting assumption of a deeply metaphysical nature -- whose veracity Einstein’s special theory of relativity and Minkowsky’s spacetime geometry must then proceed to prove mathematically. And just here is the exact point where the accusation of "mathematical illiteracy" described by Nordenson usually comes into play, because if you dare to object that this assumption about the ontological status of time is an absurd Platonism (if you happen to stand within the ranks of those who -- as Aristotle, Leibniz and, humbly, myself -- see time as nothing but the human perception of movement [1] ) you will be told that:

D) sorry, but the only possible proof for the veracity of this metaphysical assumption about the existence of time as a real entity or force and the only possible expression of this proof (verbally inexpressible [2]) are Einstein’s equations and Minkowsky’s geometry -- which, either for being as stupid as Rutherford and Poincaré or just for being as mathematically illiterate as Fabio Fernandez you won’t understand at all. [3]

Thus we are back to my question: are we really to accept meekly that mathematics can prove metaphysical propositions true (in a way that verbal logic can not)? If the answer were yes, wouldn’t become reasonable to expect mathematics to be able to prove also the existence of other Platonic Forms besides Time, and maybe the existence of God, angels, life after death and the Tooth Fairy as well?

Being not too versed neither on mathematics nor physics, I did meet many times the barrier described by Nordenson (the "mathematical illiteracy accusation") during my discussions with friends physicists whenever I tried to raise any logical objections against not only Relativity but also against any modern physical theories based on philosophic-metaphysical assumptions -- assumptions which, for my money, should be justified on philosophical grounds (therefore the problem I was ineptly wishing to allude in my previous e-mail is not really "the true nature of mathematics", which hopefully your book "What is Mathematics, Really?" will tell me what it is, but rather the true nature of many fundamental starting assertions [sometimes disguised as conclusions] in physics which being indeed of a sheer philosophical nature but unjustifiable on philosophical grounds are frequently alleged, when their philosophical unjustifiability is pointed out, to be of a sheer mathematical nature and justifiable only on mathematical grounds -- and that reminds me very much of the usual elusive moves of besieged politicians). This barrier raised by the "mathematical illiteracy accusation" would surely drive me to deep frustration would I take too seriously the alleged capability of mathematics to prove or disprove metaphysical propositions, because as the majority of mortals, who can't do what they most love to earn their livelihood, probably I will never have spare time to learn enough mathematics as to be able to understand Einstein’s and Minkowsky’s maths (suggestively, I have no trouble with quantum mechanics’ weirdness, because I can accept on philosophical grounds its philosophical implications, foundations, assumptions and conclusions without any need to resort to arcane mathematics).

Nevertheless . . . as this little self-assurance we mathematical illiterates painstakingly build through lines of thought similar to mine above very frequently reveals to be a pathetically meagre shield against the powerful doubts that mathematical savants can easily instil in the vulnerable mind of the ignorant, it would be very nice for us mortals to see a mathematician of your calibre dispelling the mists of the amazing mathematics’ metaphysical powers. So, I’m again looking forward to hearing from you . . .

Sincerely yours,

Fabio Fernandez

Note 1) I agree with Aristotle, Leibniz, Julian Barbour and many others in that time is nothing but the human perception of movement (or change) and therefore movement is the real physical phenomenon, not time. More precisely, the simple and brute fact is that mass-energy (at least as it appears phenomenally to human consciousness) is movement and vice versa: movement is not only the mode of existence of mass-energy (mass-energy "has no other option") but it is its very essence, and it is only in matter that movement exists. To say that "time flows" is the same as to say that "movement moves" (it’s not time that which flows, it’s matter; it’s not movement that which moves, it’s matter); to say that time can slow down or speed up is not only, bizarrely enough, the same as to say that "time can flow at one, half or two seconds per second", but the same as to say that "movement can move faster or slower"; it is, ultimately, the same as to say, very much in the manner of the old vitalist theory in biology, that there exist a Platonic entity, a "force" called Time which could more appropriately be called "Flux Force" and which is not only responsible for movement and change but also flows itself; so it turns out that it is this Flux Force, capable itself of flowing slower or faster, what moves the clock’s hands, in the same way that the Vital Force livens dead matter and the Locomotive Force moves the locomotive.

Note 2) Incidentally, this is also the usual argument line against objections to Hawking’s idea about "the beginning of time".

Note 3) By the way, despite my maths and physics limitations I have some learned company in feeling troubled by apparently logical aberrations in Einstein’s Special Relativity: if you feel interested in the subject, please visit http://wbabin.net/paper.htm, where some people who know much more maths and physics than me share similar worries about Special Relativity; or either read a David Deutsch’s interview at http://www.qubit.org/people/david/Articles/PhilosophyNow.html in which he talks a little about "passive acceptance" of anti-logical assertions in the physical community all over the 20th century).

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Dear Fabio Fernandez,

Thank you for an interesting and challenging letter. I am afraid that when it comes to the philosophical foundations of relativity I am out of my depth. I think your qualms are reasonable, and I hope you can find someone better qualified than I am to discuss them with you.

Sincerely,

Reuben Hersh