Limits of mathematics in cosmology

Just in case my remarks about number grammar are not clear, here is more analysis. Using Peano functions 0 and succ, there is only one representation of each number:

so 4 = succ(succ(succ(succ(0))).

I suspect that is what you mean by saying that there is no history to the numbers in Peano arithmetic.

However if we also introduce the Peano binary relation Plus, then we now have multiple methods of representing 4

4 = plus (succ(succ(0)),succ(succ(0))),

4 = plus (succ(0), succ(succ(succ(0)))), etc

These multiple representations of 4 (as they now are) have what physicists might call a "gauge freedom". The canonical representation is:

4 = succ(succ(succ(succ(0))))

This gauge freedom could be used for something, and it seems to me that ETS theory is using a similar idea to represent temporarily how the 4 got formed from its components.

"Does the struct formalism use syntactic structure (ie grammar) to encode temporal information?"

Roy, the answer is mainly "no".

You, for no fault of your own, are missing the issue of *representation*, the no man's scientific land. ;-)

In particular, a string is not a good representation because it does not contain its own formative history, and since there are exponentially many possibilities such formative history cannot be effectively recovered. (See also my paper.)

On the other hand, the issue of structural/temporal representation cannot be properly addressed outside the general scientific context, i.e.as it will be used in physics, for example.

In that sense, as a bonus suggesting the right direction, ETS possesses one critical property: its syntax and semantics are "congruent" (see my paper "Representational Formalism in Which Syntax and Semantics Are Congruent: Towards the Resolution of Searle's Chinese Room Challenge", which was written for cognitive scientists).

What this effectively means is that ETS representation of a process/object should be a direct copy of what is actually going on in nature, if the concept of structured event is indeed the right basis of the language of nature (and not triangles, circles, etc. as Galileo thought).

"Does the struct formalism use syntactic structure (ie grammar) to encode temporal information?"

To elaborate a bit more, I should mention that a struct simply records a stream of (observed) events, but it is postulated that the generation of such a stream is guided by the corresponding generative structure (class representation, a generalization of the grammar concept).

Even Stephen Hawking, in a moment of honest revelation ("The End of Physics"), acknowledged that any so-called final theory in cosmology cannot be proven to be ultimately the case, or it could always be eclipsed by new information. Why?

Godel's theorem.

In fact, this theorem places an unassailable boundary on any non-trivial mathematical axiom, etc., like those involving cosmology.

Many insightful and quasi-ultimate theories are still possible, but Godel's theorem puts a limit on mathematics that honest mathematicians and physicists would be wise to acknowledge.

Of course, accepting the reality of Godel's theorem also brings back the idea of contingency involving the universe, and since this is a philosophical concept that brings up the possibility of something outside the universe responsible for its creation, such honesty is not expected to come from those intent on disproving such a creation.

For those who cannot handle the limitation imposed by Godel's theorem, the best thing to do is ignore it as if it, too, doesn't exist. The second best thing to do is to deny the application of the theorem that even Hawking admitted applies to theories of physics. The third best thing to do is to simply claim that the theorem itself is limited to minor applications that Hawking did not realize.

There are indeed many ways to put your heads into the sand when confronted with a reality you just don't like.

But more is to be gained by just a little bit of humility as all good scientists should have, especially when trying to come up with ultimate mathematical theories that will never be and can never be completely ultimate.

DH

The ultimate limit of mathematics in cosmology is spelled out quite clearly by Gödel's theorem that even Stephen Hawking acknowledged in 2002.

Alas, it appears that only a small minority of honest physicists and mathematicians appreciate the ramifications of Gödel's theorem, which sets forth that no non-trivial system of arithmetic propositions can have proof of consistency within itself. As Hawking finally recognized (See his Gödel and the End of Physics speech/paper, 2002.), Gödel's theorem definitely applies to physics and the vain quest for an absolutely final TOE based on higher mathematics.

Hawking was, however, slightly in error in his "end of physics" lamentation. A total theory of all is still possible, but it will never contain the proof of it being necessarily true and absolutely final within itself since Gödel's theorem shows this to be impossible. Something else could always come into play to compel a change in a theory that could hold for thousands or even billions of years.

If all physicists/cosmologists would be a bit more honest as Hawking was(at least in 2002), greater insights into the nature of the universe and its creation/contingency would be readily apparent and open to greater discoveries, but even Hawking's latest collaborator Mlodinow simply dismisses Gödel's theorem as "not applicable to a TOE" (of course, he offers no evidence to support his bogus disdain, which is quite telling) in direct contradiction to what the theorem demonstrates, plus he ignores what his more accomplished co-author admitted about the theorem.

Such ignoring is definitely a most pernicious form of "ignor-ance" that does not want to accept the truth if it goes against an ideology of belief that should be absent in legitimate scientific inquiry.

So, to understand any limits of mathematics in cosmology, start with the ultimate (how ironic) limit as demonstrated by Gödel's Theorem.

Onward!

Dear All,

Everything in the universe is connected eternally in singularity, separability is an illusionary perception when one is in duality.

Relativity is the theory that best describes duality, where as singualrity is the absolute truth and several quantum mechanic observations are closer to this truth.

who am I? I am dualilty, I is the singualrity.

Please see the mathematical equation representing the absolute truth of

zero = i = infinity

Love,

Sridattadev.

If 0 x 0 = 0 is true, then 0 / 0 = 0 is also true

If 0 x 1 = 0 is true, then 0 / 0 = 1 is also true

If 0 x 2 = 0 is true, then 0 / 0 = 2 is also true

If 0 x i = 0 is true, then 0 / 0 = i is also true

If 0 x ~ = 0 is true, then 0 / 0 = ~ is also true

It seems that mathematics, the universal language, is also pointing to the absolute truth that 0 = 1 = 2 = i = ~, where "i" can be any number from zero to infinity. We have been looking at only first half of the if true statements in the relative world. As we can see it is not complete with out the then true statements whic are equally true. As all numbers are equal mathematically, so is all creation equal "absolutely".

This proves that 0 = i = ~ or in words "absolutely" nothing = "relatively" everything or everything is absolutely equal. Singularity is not only relative infinity but also absolute equality. There is only one singularity or infinity in the relativistic universe and there is only singularity or equality in the absolute universe and we are all in it.

Love,

Sridattadev

Any (mathematical) cosmology theory that posits the a priori existence of highly complex mathematical structures prior to the creation of the universe is implicitly positing the existence of a mathematical world which is eternal. There are not theory of creation of the whole universe but instead explain how the mathematical eternal part of the universe at some point rules the evolution of our world from it. Nothing explain the existence of this relatively complex mathematical core, and nothing explain the connections between the mathematical world and reality.

Hi Lev,

In the 1990's I developed a theory of visual perception (Ph.D thesis: Perception of the image world") that developed a concept of image structure hierarchy corresponding to the ontogeny/phylogeny of objects.

"In light of this, the 'similarity' of objects should be understood as the 'similarity' of their formative

histories, and hence this formative history (or object's structure) must be captured in the object's

formal representation."

In my approach, the detection process of an image structure recapitulate in its sequence and structure the morphogenesis process. This is a modern version of the old aristotlean conception of the actualization of the object's form into human mind at it intellectual level. The platonic world in this model in implicitly built-in in the hiearchical structure of the vision system of animals.

Postulate 1: the universe is a family of evolving and interactive classes of (irreversible)

processes.

I would prefer:the universe is an evolving family of evolving and interactive classes of processes.

Notice that with the structural hiearchical representation, it is not necessary to represent time by a parameter, or a dimension of space. By the universal principle that STRUCTURAL HIERARCHY IS HISTORY, you get a Bergsonian duration built-in.

- Louis

All of mathematics is limited by Axiom of Choice. I propose that Axiom of Choice can be extended to include Relativity, and therefore define a model by which ALL of observable physics can be modeled.

Axiom of Choice extended to include Relativity

This model potentially describes the foundations under quantum entanglement and related decoherence, time independent photon interactions within fringe patterns, entropy, duality, time, gravity, subatomic particles, deformable space, ...

This is only a foundation for a cosmological model based upon quantum causality that allows for building evolving relativity upon a base of causal states. The system has the potential to model diverse physics environments different then our own; but our own as one set of non-relativistic singularities that moderate relativistic singularities (sub-atomic particles).

The remainder of the thought mapping that has accumulated is found at:

QESdunn

Until corruption can largely be eliminated, I do not want space-time manipulation to be developed. The reason is that then weapons will otherwise be developed to enslave the masses. Manipulating space-time allows for remote manipulation of subatomic particles. Which with advanced application can mean automatically detecting distraction, and causing pain. Not just in self, but in all those whom you care about. Ultimately causing death if acting outside of boundaries.

Eliminate ALL Corruption

retweet: Part of Civil Rights is that Representation is free of Treason http://tinyurl.com/lpqsur5

My theory on the limits of mathematics:

Theory of Everything using Leibniz, Kant and German Idealism

James,

The Axiom of Choice may be replaced by an Axiom of Exclusion that isolates a Natural Index of 12 numbers that index (by dimension) the mathematical structures encompassing physics. Incorporating this into proof theory leads to a proof of the Riemann Hypothesis. Of course, to get the $1,000,000 from the Clay Mathematics Institute requires that you also convince them that you are allowed to change the rules of proof.

My theory on the limits of mathematics:

http://www.academia.edu/8991727/Phenomenal_World_as_an_Output_of_Cognitive_Quantum_Grid_Theory_of_Everything_using_Leibniz_Kant_and_German_Idealism

Ramin Zahedi,

You may be looking forward to criticism/ comments on your work. But I think this will not be by multiple postings on different forum on this site. I have taken a look at your article. What are your fundamental postulates and assumptions? What transformation equations are operational: Galilean or Lorentzian?

My initial impression is that only three people in the world can understand this work. If you want to discuss it seriously, you may wish to invite a dialogue on a forum such as Alternative Models of Reality or other similar forum on this FQXi website.

Sir J.J. Thomson, as President of the Royal Society at that time, concluded the 6 November 1919 meeting of of the Royal Society with the statement, "I have to confess that no one has yet succeeded in stating in clear language what the theory of Einstein's really is". And Eddington recalled that as the meeting was dispersing, Ludwig Silberstein (the author of one of the early books on relativity), came up to him and said, "Professor Eddington, you must be one of three persons in the world who understands general relativity". On Eddington demurring to this statement, Silberstein responded, "Don't be modest Eddington". And Eddington's reply was, "On the contrary, I am trying to think who the third person is!"

Up till today, only this two or three can still claim knowing what this theory is and I don't think you are included or are you?

Akinbo

Hello Ramin,

I didn't get your email and I am not on msn. I would prefer that the discussion be public rather than private so others can contribute or correct whatever is wrong in our posts.

Some comments from James Putnam and myself are waiting for you on the Alternative Models forum

Regards,

Akinbo