Essay Abstract

Pure Physics is not Pure Mathematics. In comparison with mathematicians, physicists agree to simplify mathematics because in some cases it " naturally" arises from physical experience. Hence,some basic mathematical assumptions are defined as Wrong ( or not sufficiently realistic ) statements in physical sciences. In this context we can develop even the whole comparative theory of physical - mathematical differences from this point of view. In order to show this kind of anthropic comparative studies let us consider merely two the most sensitive passages of Relativistic Physics. Passage 1: Einstein physical theory loci in space. Passage 2: Einstein's Time as a new kind of complex number.

Author Bio

Oxford based anthropologist and cryptanalist ( quantum immune cryptography )

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Dear Michael,

I read your essay; the relation between physics and mathematics, especially at the foundational level, is a very interesting topic. Your essay raises some questions, though.

First a comment: on page 2, you wrote: "In Pure Mathematics real number may be regarded as the measure of a displacement." Actually, no. In pure mathematics, a real number is usually regarded as a so-called Dedekind cut, which is an intersection of a (possibly infinite) number of intervals. There are other definitions as well, but the point is that in pure mathematics one does't use measures or displacements to define real numbers. It is rather the other way around: one uses real numbers to define a measure, which usually can be seen as a function from some set to the real numbers.

Then some questions.

1) You write that "Einstein's equation s² = x² * y² * z² is able to produce some unsolved number-theoretical problems" (I used the *-symbol for addition). As an example, you state that for odd integers x and y, no integers s and z exist that satisfy the equation. But why is that important? The terms in the equation, to which you refer as Einstein's equation (by the way: why?), are namely real numbers, not integers. That means that for any real numbers x, y, and z there is always real number s that satisfies the equation. That real number s doesn't have to be an integer. So where is the problem?

2) You write that "a displacement [ x', x'', x''', y ] is simply another way to say that there exists a kind of ( x' * x'' * x''' * yi ) complex number, or a new kind of complex number" (again with the *-symbol for addition). The question is: why do you need a new kind of complex number? The point is that a spatiotemporal displacement < x, y, z, t > can perfectly be represented by an element (vector) of the set R4 = RxRxRxR (where R is the set of real numbers). Of course one can also represent the same spatiotemporal displacement by a quaternion t * x.i * y.j * z.k, where a quaternion is (loosely speaking) a four-dimensional complex number. But there is no need to do that. All calculations, that can be done with the quaternions, can also be done with vectors. So why is any of this related to a wrong mathematical assumption in physics, the topic of your essay?

Best regards, Marcoen

Dear Vladimir,

Thank You. I suppose that human research tradition based on entangled co-existence of Mathematics and Physics has sufficient foundation and very long history.Practical wisdom or Philosophy of such sort of evolving bi- tradition is open problem,connected with foundations of mathematics,indeed.

However, we may await that any future X-ontology cannot change nature of human mathematics as well as nature of human physics.

Physics is Physics, and Mathematics is Mathematics.

Dear Marcoen,

Thank You for questions.

1. Real numbers MAY Be regarded as the measure of a length, it May Be regarded as the mark of a point and it May be regarded as the measure of a displacement or change of position on the line ( complex numbers case ).

2.Indeed, each of real numbers can be represented in terms of a ( signed ) infinite dicimal expansion. However, when Einstein uses Pythagorean exactness in his statements in pure physics, we can make comparison with Pure Mathematics.36^2 8^2 3^2 = 37^2, but there is no such odd^2 odd^2 z^2 = s^2 even in nature. In order to represent the same idea in real numbers you simply must change the sense and a form of equation.

As is known, Pythagoras theorem is basic valid assumption of Relativity.

3.Idea of a new kind of complex number (inspired by Einstein's physics) was developed by R.Penrose in his theory of twistors.The basic twistor is four -complex - dimensional object of the type Z# = ( Z, Z',Z'',Z''') where Zs are four complex components.Following Penrose, the twistor co - ordinates Z,Z',Z'',Z''' together with their complex conjugates ( vectors) Z,Z',Z'',Z''' can be used in place of the usual Einstein x,y,z,t and their canonical conjugates. In fact, anything that can be written in normal Minkowsky spacetime can be written in the terms of twistors.

My comparison with pure math of complex numbers may suggest that it cannot satisfy pure mathematical justification of a new kind of complex numbers.Moreover,there is pure mathematical Pontrigain theorem on limited generalizations of complex numbers which could be used also as argument.

best regards, Michael

a month later

Dear Hoang,

If I understand your questions you try to connect Einstein's equivalentness ( of inertial and gravitational mass ) with Higgs boson by some kind of philosophical definitions. I suppose that mathematically speaking, however, there is some technical problem here, connected with "unknownness" of Higgs-like complex scalar algebra...?

" No fundamental scalar fields have been observed in nature..."?

1. Higgs boson existence.

I suppose an existence of mathematical object ( called Higgs Boson ) as a consequence of

Conjecture( Goldstone - Higgs ( 1964 ))" Loretz - covariant field theories in which spontaneous breakdown of symmetry under an internal Lie group occurs contain zero-mass particle fields iff the conserved currents associated with the internal group are coupled to gauge fields ".

But, I cannot accept Higgs' simplification of true algebra of complex scalar in particle physics.Taking seriously.

2." Mathematics is just a tool".

I think, it is very popular and very naive error. Pure Mathematics is area of experimental science. Anybody can test that x^3 +y^3 = z^3 cannot exist in Nature( please test it, may be you can find counter-example,because following your assumption " numbers are merely abstractions","human subjectivity",etc) If math is "only language" you always can re-write such theorem in "suitable solvable" manner and you always can "find" solutions for any mathematical problem reduced to language paradox. Similar with Pythagoras theorem used by Einstein.If Math is just language,you always can re-write such sort of theorem for any numbers. But, without Pythagoras theorem Einstein theory cannot be formulated at all. Please test it.

Cao Hai,

Your statement "Mathematics is the "logical inference" of who to use of mathematics, that is "conscious of Physics " of us " is assuming that mathematics is something always and everwhere reducible to logical operations only due to logical axioms. However, if it is correct, we always can change logical system of axioms in order to prove that x^3 +y^3 = z^3 (I suggest that this consequence of Fermat theorem is impossible in any logical system of axioms as well as in Nature)because Nature has own mathematics and own "lovely" representations.It is easy to test in physical laboratory.

Generally,in contrast with popular philisophazing physics, Mathematics and Physics are different kinds of Experimental sciences used Computational and Physical experiments to understand Nature.

This means if you are physicist, it is natural to await that you can offer EXPERIMENT, but not philosophical hypothesis, in order to prove your version of the wrong physical assumption,indeed. Unfortunately, I am sorry, but I cannot find any manifestation of experimental physical thinking in the contest...

Dear Michael,

As promised, I have just finished reading your essay! Quite interesting. A couple of thoughts:

1. It's certainly true (at least, in my opinion!) that not all mathematical constructs are useful or appropriate for describing physics, and it's also true that physicists have invented (or discovered, depending on your point of view) a lot of new mathematical ideas in an attempt to describe physics.

2. There are many connections between complex numbers and geometric ideas in spacetime physics (I see now that my answer to you over on my thread was mostly irrelevant to the point you were making). For example, special relativity in two dimensions can be described entirely in terms of complex numbers in the complex plane rather than two-dimensional Minkowski space.

As you point out, the fact that a minus sign appears in the last term of the expression describing the square of the relativistic displacement does associate an imaginary factor with the time variable.

In four-dimensional spacetime, there are also several other ways to think about complex numbers or generalizations of complex numbers. Quaternions, for instance, are generalization of the complex numbers in which there is one real unit 1 and three imaginary units i,j, and k, all of which are different but all of which have -1 as their square. Here is an article about expressing special relativity in terms of quaternions.

Another somewhat similar idea is Roger Penrose's twistor theory. In "twistor space," there are two plus signs and two minus signs rather than three plus signs and one minus sign as in Minkowski space.

Anyway, I thought you might find that interesting. Thanks for commenting on my thread. Take care,

Ben

If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is [math]R_1 [/math] and [math]N_1 [/math] was the quantity of people which gave you ratings. Then you have [math]S_1=R_1 N_1 [/math] of points. After it anyone give you [math]dS [/math] of points so you have [math]S_2=S_1+ dS [/math] of points and [math]N_2=N_1+1 [/math] is the common quantity of the people which gave you ratings. At the same time you will have [math]S_2=R_2 N_2 [/math] of points. From here, if you want to be R2 > R1 there must be: [math]S_2/ N_2>S_1/ N_1 [/math] or [math] (S_1+ dS) / (N_1+1) >S_1/ N_1 [/math] or [math] dS >S_1/ N_1 =R_1[/math] In other words if you want to increase rating of anyone you must give him more points [math]dS [/math] then the participant`s rating [math]R_1 [/math] was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.

Sergey Fedosin