Heuristic rule for constructing physics axiomatization by Florin Moldoveanu

Dear Florin,

thanks again for replying.

I have an analogy to contribute concerning the platonic realm, infinity and my own asumptions about "empty space".

Imagine there exists nothing. Nothing but an (in)finite (white) plain. This is my "empty space".

Now, imagine drawing a circle onto the plain. The plain has now a circle on it. Was the circle there at the plain before "someone" has drawn its borders onto the plain? Surely, and i agree with you.

Now imagine not drawing a circle onto the plain, but a rubber duck. Was the rubber duck there at the plain before "someone" has drawn its borders onto the plain? Surely, i would say, because i know what Rubber duck looks like and what it means and can imagine it to be on the plain.

The meaning of a circle in our universe is fascinating. It is symmetric, a strong symbol and a very practical thing too. But it is anthropic (in the sense of being used to use maths as everyday-language) in my opinion to think that the whole plain contains nothing more but mathematical relationships. I will try to expose this.

There may exist relations that are immovable. Pi could be one of them because the too properties of a circle (diameter and Perimeter) that relate to pi cannot be considered as independent from each other. Because a circle is a "oneness". If we would divide the circle into two halfs, we couldn't obtain anymore pi. So pi is a property of a categorial oneness, named "circle". The same is true for the rubber duck and for the number "e". Whenever i imagine this rubber duck in the same way i did it in the example above, i get the same result. And the same is also true for the imagination of the whole landscape of maths. Whenever someone speaks of the whole landscape of maths, he/she presupposes a certain picture of it.

Although we can describe my rubber duck very precisly by mathematics (for example by a computer animation), two things we can't describe by maths: Firstly my inner impressions about the colors red, green etc. and many more things that are subjective. And secondly also we can't describe by maths the whole landscape of maths itself. At least i do not believe that we can do that. My conclusion out of that is that (out of space-time) there must exist some more than only maths. Because math may not need space-time to exist, but it needs a oneness that it can divide into parts and relate them to each other. So maths is not an independent concept that can exist on its own right somewhere, but depends on something more.

The term zitterbewgung is German for frantic motion, which goes back to the very early days of quantum mechanics. Tehre are those who interpret this as a hidden variable, which I don't do. Zitterbewegung does suggest some sort of confining of a particle by a potential field however.

LC

Dear Florin,

thank you for your feedback and your effort to make me understand your point of view. I am thinking about your comments and contemplating the possibility of your arguments being right, absolutely right or partially right. Please let me therefore make a few final remarks.

Concerning our main theme here, the question of what could/couldn't be possible in physics, although it may be provable that the maths underlying our actual universe is consistent and coherent and other maths wouldn't, this is in my opinion not sufficient to conclude an a priori platonic realm of fixed axioms and their consequences and hence the assumption that outside space and time there exists nothing but pure maths as an abstract concept.

You wrote:

"I do not understand the oneness idea, but again I am not a philosopher."

The oneness-idea is simple and i tried to visualize it by my example with the white plain. Even if one defines borders on it, it stays the same regarding its essence and contents.

So let's leave aside my white plain, for it is only an attempt to give a helping construction to illustrate. Some mathematical values may or may not exist forever on this plane or elsewhere, that's not the main point i am concerned with. My question is, if the whole axioms that are generally possible by reasoning (in maths) exist a priori in some platonic realm or not.

You wrote:

"Reasoning and math are not ill defined in the absence of mathematicians"

When understood as absence of everything but abstract math, i would doubt this. Because the main axiom presupposed in the whole discussion is that maths is rather incomplete then inconsistent. There can be many axiomatic systems that are at odds with each other and lead to mutually exclusive perspectives about the validity of the arguments. All that is surely nonetheless based on reasoning and therefore resoning isn't an ill-defined concept in that case. But the conclusions drawn out of these maths could be simply arbitrary and therefore not even ill-defined.

I think we both can exclude not only for the purpose of the arguments, the case that reasoning is a priori deeply inconsistent.

But according to Goedel, every thing that can be considered as absolute, complete and definite, must contain itself as an element (so is our body, which is coded in detail in our brains). And to bind together a mathematicians or someone else's world of ideas (into a class of all the sets of thoughts and conclusions) to a coherent unity, this world must be an element of itself and can only be grasped by irrational insights. Because no rational thought is an element of itself. And therefore no rational thought can bind together one's world of ideas to a unity. The question is if it is rational to consider a completely fixed and differenciated landscape of maths to be preexistent without knowing if this landscape is fully coherent, non-random and well-defined right from the start. I think this question is not yet decided (and maybe cannot decided here and now), but i don't exclude the possibility that it can be decided in the future by a strict proof that convinces every practitioner at the end of the day.

Dear Florin,

thanks also for your feedback!

you wrote:

"In other words, just because axiomatic systems are at odds with each other, this does not make them ill defined."

That's not what i am saying.

I said:

"There can be many axiomatic systems that are at odds with each other and lead to mutually exclusive perspectives about the validity of the arguments. All that is surely nonetheless based on reasoning and therefore resoning isn't an ill-defined concept in that case. But the conclusions drawn out of these maths could be simply arbitrary and therefore not even ill-defined."

Firstly, i do not consider Peano or Pressburger arithmetic to be ill-defined. They are just defined (on the grounds of reasoning). In the case of the Pressburger arithmetic, Gödel's restrictions may not hold anymore. So we have to systems, Peano arithmetic and Pressburger arithmetic. The first may run on a PC, the latter may run on a Mac. But to conclude out of that with necessity that there must exist a(n) (infinite) platonic realm of all abstract maths with all possible and thinkable axioms already fixed and independent of perception, that's what i doubt and what i tried to say.

You wrote:

"Reasoning and math are not ill defined in the absence of mathematicians...and this has nothing to do with Gödel's theorem."

Yes to the first part of your statement concerning reasoning. No to the statement concerning "ill defined", because in the absence of mathematicians, i don't think that there exist fixed mathematical axioms independent of human reasoning. And no to the second part of your statement, because Gödel's theorem is a result of math and (human) reasoning and for Gödel's result i don't believe that it could be an everlasting, eternal truth in a platonic realm of abstract maths. Not because i think that "independent" axiomatic systems are deeply contradictory, but because i think that there isn't such a fixed platonic landscape of all possible ("thinkable") mathematical axioms, theorems and finally distinctions.

Godel's thoerem says that no axiomatic system is able to enumerate all of its own Godel numbers. The Turing machine approach I find more understandable. A Turing machine either halts or it does not. A Turing machine can get caught in a loop, such as MicroSoft's software often does. So it would be nice to have a universal Turing machine which can determine whether any Turing machine will halt or not. So the UTN emulates all other TMs, including itslef. This leads to a problem since the UTM must emulate itself emulating other TMs, and so forth. This leads to something isomorphic to the Cantor diagonalization argument with Godel numbers: No axiomatic structure can enumerate or encode all its Godel numbers. This means consistency is not provable in any formal axiomatic system.

This inconvenient fact has strangely had minimal impact on mathematics so far. Most proofs, including the outsanding proofs of conjectures by Fermat and Poincare, are not impacted much by this. Though there are some curious aspects with this involving "proof theory."

LC

Dear Florin,

thanks for your feedback too.

If i would believe in solipsism, i wouldn't correspond with you, because there wouldn't be someone to correspond to - who has feelings and is much more trained in logics than i am.

But nonethelesss, i don't understand your conclusion of solipsism out of my writings (maybe you could expose this more in detail). All i said about my belief is, that i don't believe that there is such a thing as a fixed infinite landscape of mathematical axioms, theorems and mathematical inductions. So being sceptic about this, i would say i am merely a skeptic (and maybe a displeasing one, - i would admit :-).

And maybe i am wrong in my belief that there isn't such a thing as the fixed mathematical landscape. Maybe it does really exist!! (for whatever reasons). For that case i would not automatically assume that that's all that can exist. Although the imagination of a fixed infinite landscape gives the impression to me that there isn't much "room" anymore for other things to exist because this landscape would be somewhat "infinite in all directions", i am not convinced of the conclusion that in the case of the existing fixed infinite landscape of maths there couldn't exist additional features of ultimate reality.

Provability, i think, doesn't lead us any further here, and also the term of truth. It seems to me that an intuitive description of ultimate reality must at first place have an overall consistence that, for all practical purposes in the universe we live in, at least seems to be the ultimate truth. Maybe that's the case with your fixed landscape and your universal truth property in cooperation. Therefore i insist on my comment from Aug. 4, 2009 @ 07:36 GMT, second chapter. My comments on your essay and your concept were thought as comments on possible loopholes that one could consider as relevant regarding the results you may obtain with your approach. But sorry for my somewhat smart-alecky type of criticism. It is my type to consider radical, new and far reaching assumptions about the world we live in, because such assumptions could have - and in my opinion indeed have - impacts on our view of the world (at the latest then when they are presented in popular scientific magazines especially also for people which can't evaluate every line of reasoning in every such assumption). I am convinced that this is a legitimate behaviour in a forum like ours.

Dear Florin,

i am not confused. Because "solipsism" is too crazy for me to waste more than the one or two following thoughts on it for the aim of considering it as deeply irrational. Everything else would be deeply irrational, wouldn't it?

See for example the possibilities of a "solipsist" to explain the only fact (in the meaning of a fact that does *resist* until this solipsist dies) that he can state about his world: The fact that he or she is the person he/she is and not another person with another body.

In a solipsist's world, there is no explanation for the fact, that he/she is indeed he/she and not - his aunt Daisy. To circumvent this question - the question why he not looks out "into" his/her solipsistic world through the eyes of his/her aunt Daisy and doesn't exhibit the aunt's body but all of things has his/her own body day by day -, the only way for him/her is to *deny* flatly the consistent explanation that exists in his own world: Namely evolutionary and historical accumulation of events. So to be a "real" solipsist, one has to *deny* scientific explanations (*and* therefore the fact that maths *can* predict many things in our world) or having not yet heard of those explanations or don't having very much memory at all. I think i haven't created the impression on FQXi that i do not believe in both - the ability of maths to predict things in our world and the fact of evolutionary, historical accumulation of facts in our world.

But there may be things in the universe we don't know. Nonetheless one thing i take for guaranteed: To be able to grasp and understand these things, they must be structured. Not only structured, but hierarchically structured to be comprehended by us consciously. The same would be true for the platonic landscape of mathematics. If it is structured hierachically - then maybe it could indeed exist in my opinion and in that case would have somewhat "infinite" logical deepness (the latter i would then not call "infinite logical deepness" but "intelligence"). But if it isn't structured hierachically, it would be only an aggregation of random self-assuring tautological things and each of these things would look for us humans random in respect to it's questionable existence.

Why the latter isn't the case for Pythagoras' theorem or for Gödel's incompleteness theorem (or pi), lies in the fact those structures to a certain degree are properties (in the sense of sub-structures) of the world we live in and this world has a historical evolution that is hierarchically structured and came into existence out of certain profound reasons we yet still don't know for sure. Therefore the mentioned mathematical theorems surely existed 5 billion years ago before Earth was formed, because our universe we live in existed 5 billion years ago before Earth was formed.

So please don't push me in a direction i don't belong to: Namely intelligent design and related issues. I believe in intelligence, but i don't believe in fairy-tales.

I read Chaitan's paper some years ago. He appears to argue that mathematical systems emerge from self-referential propositions as accidents of sorts. As for the continuum there is of course the matter of the continuum hypothesis. Bernays and Cohen demonstrated by Godel's thoerem it is consitent with ZF, but not provable.

LC

Dear Florin,

I am not quite sure if i understand the meaning of your term "solipsism". I think you mean the problem that the decoherence-concept could break down if the universe is considered as causally, physically closed. Is that right? (solipsism as i heard of it is the position of an individuum to assume itself as the only individuum to exist).

I do not believe in superpositions in a mainstream-manner as real mutually exclusive states somewhat and somewhere overlapped with each other until divided by a measurement.

Superpositions for me are another term for an "undefined" state. That they look pretty like overlapping real states is due to the environment and its imposing limits, in which the undefined area gets a special contextual meaning for experimenters and their calculation-schemes.

"So let me try again to restate my understanding of your position: once the universe starts to exists, math comes into existence as well. Is this true?"

No. Once the universe starts to exist, something other than maths is transformed into dynamical structures and substructures that can partly be grasped and translated into the language of maths.