Coincidence or correlation between two separate areas of mathematics may, for pragmatic reasons, lead to a unification of the two or a treatment of one as a reduction of the other. That can be accomplished with bridging definitional identities, which are stipulated. Definitions are tautologies, and tautologousness would be inherited by many derivations from those definitions.

Actually

sin(x) sin(x) +cos(x) cos(x) =1 is Pythagoras in disguise, so it actually means something. Many identities seemingly dont.

Wittgenstein thinks math is a huge tautology. Is this exaggerated?

If math is only built on definitins, then he would be right.

However most of his discussion does not seem to lead anywhere.

I agree. The misunderstanding of equality (only the numbers are equal) is a great cause of bad physics. For example, those suggestions that wave characteristics such as frequency is related to particle characteristics such as momentum.

Physics use of mathematics

http://intellectualarchive.com/?link=item&id=2557

John

The relation you are talking about is in the science category, basically empiric, so it does have significance.

It reads

p= h/lambda

p magnitude of particles linear momentum

h Plancks constant

lambda Greek letter denoting wavelength(of a wave). The particles wavefunction.

This is called de Broglie relationship.

Physics uses math but other things as well(ie experiments)

This kind of equality differs from a purely mathematical one because the

quantities have physical units., and physical interpretation.

Also physical quantities may be subject to things like measurement error.

Not quite so literal as math usage.

In this case care must be exercised in interpreting the formula. Saying that the wave and particle are the same thing is wrong.

The wave is not the particle, but its important in the probability of finding the particle.

Total tautologousness at the mythical "end of inquiry" may be a goal at which mathematics aims, but I venture there will always be areas of mathematics that harbor inconsistencies that may never be revealed or metamathemetical statements whose status as necessary truths will remain unresolved.

Karl

Is not that a bit extreme? I dont know any branch of inquery that wishes to suicide itself.

Agree that it is the source of discussion, that will not end anytime soon.

People still claim they dont know what its about, but if they know, disagree with others.

I still atempt to keep a positive attitude.

Juan Weisz

I dunno. Mathematics is an infinitely large palimpsest, whereas human lives (and ∴ the life of our entire species) are finite. We can't hope to make more than small areas legible and we'll make mistakes in the process. I don't understand why you think this admission constitutes suicide. Even if it turns out sub specie aeternitatis that much of our mathematics is merely heuristic and satisficing, inasmuch as we don't have access to the whole parchment, there is surely always enough unfinished business to keep busy everyone who wants to be. Besides, even inconsistencies might be the impetus to venturing into previously unknown mathematical territory. Mathematica longa, vita brevis.

Karl

Well, as a retired physicist, you think that some statements you can make, or have made, are more meaningfull that the mere thought that something must be equal (or identical) to itself.

I dont think math can be entirely reduced to a tautology. It is likely that parts of it can.

It souldent satisfy a philosopher either? Similar problems with that?

Well some in math say that its just a form of addiction, others an art form, others a science.

Well, as time passes, things become different also, as they say, you never bathe

twice in the same river.

Check https://www.researchgate.net/publication/279514041_Equality_and_identity

The concept of coincidence as opposed to identity can be illustrated by

assuming that a is a number drawn from a box A and b a number drawn from box B

Suppose box A contains numbers 1 and 2, and so does box B

Then a=b is true if the same number is drawn from each box, but this is true only half the time.

Now if A contains all the real numbers and so does B, then the result a=b is a miracle, for random drawing of one number from each box.(strongly suggesting it is not miracle, but has some reason)

So identities have something to do with probabilities?

Juan Weisz

A known identity has a probability of 1. Lesser probabilities are a measure of what we don't know, not what is. Drawing the particular result is an unpredictable occurrence that we had no reason to expect.

«(The number drawn from box A at t) = (The number drawn from box B at t)» is always true (or false) for a particular value of t. To simply speak of a = b is misleading, since your example is sensitive to time.

You say: "Then a=b is true if the same number is drawn from each box, but this is true only half the time." Why? The prior probability may be 50% but it doesn't follow that the world will provide 50% compliance.

And BTW: You can bathe in the same river twice, just not the same water. It's like a bathtub except the water you bathe in constantly changes.

Check this paper also

https://arts-sciences.und.edu/academics/math/_files/docs/courses/208text/math208discretemathv2-1.pdf

Karl

The way I read it is that the possible events are

(11)

(12)

(21)

(22)

and favorable events

(11)

(22)

Here Im dealing with coincidence, not identity. They are not the same,

need more info for coincidence.(ie define sets)

For identitythe extra info may be the rules of algebra (or arithmetic)

I would not use this analysis for identity.

Regards, Juan

Empiric probability is supposed to be the same as a priori probability (to give same results, with error bars)after many trials.

Kiprotich, interesting paper to learn from.

(though that paper is faily conventional discrete math)

Another meaning of equality, may be accidental.

Dear Juan Weisz

With respect to "identity", it is an unresolved problem in the philosophy of logic whether there is such a thing as the logical principle of identity, and much has been discussed on the subject.

One aspect of the matter is that identity is not conceived -or defined- as a property of an entity but as a relation of a thing to itself, but the problem is that it is usually considered that for a relation to exist there must be at least two relata or terms of the relation, and in this case there would be a type of relation in which there are not two terms but only one, to which authors like the early Wittgenstein have objected, since if it is held that there are relations in which there are not at least two related things it is no longer possible to differentiate a property from a relation.

Indeed, perhaps this is so, as an argument by Francis Bradley in Appearance and Reality points out, which holds that the concepts of property and relation are mutually presupposed so that, individually taken, neither is "intelligible."

A Carnapian nominalist might say that the word "relation" is used with another meaning, giving rise to a meaninglessness that is the product of a lax use of language; from a logical perspective, one might say that it gives rise to contradiction (and that is why more than one logician has not accepted it as a postulate or principle of logic).

Likewise, the so-called "identity paradox" has been precisely what led Frege to make the distinction between sense and reference, which suggests that there are deep philosophical problems in assuming that there is a logical principle of identity or that there can be a relation of identity of a thing with itself.

In this respect, Wittgenstein's Tractatus says: ""Roughly speaking, to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing at all" (5.5303) "Identity of object I express by identity of sign, and not by using a sign for identity" (5.53).

Regarding "a and b could be different numbers", two things have to be remembered: that a and b are variables, physical symbols, that they represent numbers (which are not physical), and that identity is not the same as equality. With regard to "tautology", I understand that it is a term of logic and not of mathematics, although perhaps the principles of a mathematical theory can be said to be tautological, in some sense, and this will depend on the ontology that one assumes, because there are arguments that define "tautology" in a merely syntactic sense, others in semantic terms of truth values, others of possible worlds, etc., but it is something different from the presumed principle of identity or the presumed logical relation of identity of an object with itself, and also of logical equivalence. (But it is also possible to consider the principles of mathematical theories to be analytic statements and not tautologies.)