On the Hypotheses of Nothing Which Lie at the Bases of Mathematics and Their Consequences for the Foundations of Physics by Jonathan Cender

Jonathan Cender

Essay Abstract

Abstract. As is well known, re-thinking assumptions which lie at the bases of geometry about, as Riemann put it, "both the notions of space and the first principles of construction in space", led to the creation of curved, nonEuclidean geometries. These contributed in turn to re-thinking certain basic physical assumptions. I contend that by re-thinking the contrasting assumptions about nothing which lie at the bases of mathematics, particularly placeholders and the number 0, new math can be created, notably a number zero more useful for physicists. Once again, the way will be open to re-thinking fundamental physical assumptions. The new zero, while still working exactly like 0, also happens to work in ways mostly, but not entirely, similar to "physics math" (math devised by physicists to do things they couldn't with existing math). The similarities give some cause for optimism that the alternative number of nothing and some other related math shares some basis with reality. The question then becomes "Do the dissimilarities to existing physics math indicate a place to begin questing for wrong physical assumptions?" After a brief foray into the aforementioned assumptions of nothing, an alternative zero arithmetic will be introduced with special attention to its similarities to calculus, the Dirac Delta function, the extended complex plane, and to a notation for arrays of real numbers developed by Roger Penrose and John A. Wheeler to compensate for limitations of the number concept "cardinal" when working with n-real-dimensional space. Finally, some physics math that doesn't match the new math will be covered. Examples given relate to singularities and to the 0 dimension.

Author Bio

Undergrad - A few years off and on at Harvard University. No degree. Research assistant for a number of years in the Economics Department of Stanford University and later at the Hoover Institution. Programmed econometric models of economies, primarily of the People's Republic of China and the energy sector of the United States, and later assisted in a book on tax policy. Also worked on data analysis for the first study of the use of email in the U. S. Itinerant tutor - including a brief stint with Disney's Mickey Mouse Club. Skills trainer for children with autism. Currently I reside on a lovely Pacific island.

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[deleted]

Jonathan, I enjoyed the breadth of your ideas and the clarity of your presentation. It looks to me worthwhile in many uses of math and physics to develop the idea of 0 as a void that has many possibilities. Additionally, it could play a role in physics in the more speculative issue of how consciousness would relate to presently known physical laws if it is able to produce a physical effect without any accompanying physical (deterministic or random) changes, as in the case of free will. (It isn't known whether we have free will, but I think the idea is worth investigating.) If in fact we have free will, the space describing choice could well be a void which holds latent possibilities. So the concept would be relevant to the study of this aspect of consciousness.

Jonathan Cender

Thank you for your kind words.

Your idea that choice could be void holding latent possibilities is provocative. Given your interest in free will here's a link to writings by the astrophysicist David Layzer on free will and entropy. The link is waaay at the bottom of the page along with other links on papers having to do with free will and information philosophy in general. An excerpt:

Naturalizing Libertarian Free Will

David Layzer

Department of Astronomy, Harvard University

www.informationphilosopher.com/solutions/scientists/layzer/

Libertarian free will is incompatible with the thesis that physical laws and antecedent conditions determine events other than the outcomes of quantum measurements. This thesis is not a consequence of physical laws alone. It also depends on an assumption about the conditions that define macroscopic systems (or, in some theories, the universe): the assumption that these systems (or the universe) are in definite microstates. This paper describes a theory of macroscopic initial conditions that is incompatible with this assumption.

[deleted]

Jonathan,

Very interesting essay!

You stated:""Nothing" in the placeholder sense returns us to the origins of the word zero. The Sanskrit word sunya, from which the word "zero" descends, refers to a relational, full, or pregnant void rather than to a void that is barren or empty; a void full in the sense of possibility and the potential for relationship with what is here."

This concept works very well also with my own essay, although I am not sure if it is what you had in mind.

My essay postulates that we have been anti-differentiating the Newtonian field incorrectly, which leads to the incorrect form of the Einstein field equation. If a function F1 is Newtonian field strength, then F1' is gravitational force. I state that we have probably been mistaking F1' for (C-F2)' so that the Einstein field equation with the cosmological constant should read similarly, i.e.[math]G_{\mu\nu}=\Omega g_{\mu\nu}-L_{\mu\nu}[/math]

where the constant term is equated to a potential for energy of the vacuum and Luv is equated to a residual dynamic stress energy tensor. This would allow a large magnitude for the constant while still appearing like attractive gravity but also allowing a repulsion after a certain radius. Solves the old and new cosmological constant problems.

Where I see this relating to your concept is that while a stress energy tensor for Guv may become zero or "nothing", if the two tensors of the right half become equal, the sum may become zero but that certainly does not imply it is "empty". This would very much seem to fit the full void of "sunya".

Regards,

Jeff Baugher

Jonathan Cender

Glad you found it interesting.

I'll look at your essay and see how the sunya nothing compares. Zero in the equation above does sound similar.

Jonathan Cender

Jeff, my response is on your essay site and dated October 1.

Benjamin Dribus

Jonathan,

I read your a couple of times, and enjoyed it, but I confess I don't entirely understand it. Maybe you could clarify the following points for me. I will call the Wallis zero w.

1. You say that "present arrays of the form n(infinity) only arise by division." I assume you can only mean division of the form n/w=n(infinity)?

2. If this is true then (infinity) means 1/w?

3. You distinguish "reciprocal" from "multiplicative inverse." Does this mean that the binary operation of division is not the inverse of multiplication in your "unreal number system?"

4. I don't see how 1/(infinity) is an "array number" according to your definition, yet you apply the absence bar to it. Are you allowing repeated division to define array numbers; i.e., 1/(infinity)=1/(1/w)?

By the way, I will mention that there are some other possible ways to think about "nothing" in this context. For example, suppose you have a set with a partially defined binary operation. If you consider two elements whose product is undefined, you get "nothing" in an obvious sense. Now, you can form an algebra (using any ring as coefficients), by using the set with its partial operation as exponents; a product retains all the elements whose exponents combine under the partial operation, and the empty sum is interpreted as zero. In this sense, "zero equals X^(nothing)" in a sense, so "nothing" is actually like a logarithm of zero instead of zero. This is exactly what happens for the path algebras I discuss in my essay:

On the Foundational Assumptions of Modern Physics

In any case, it would be nice to have a clean axiomatic description of your ideas. Nobody pays enough attention to objects like this even though they arise in physics. Take care,

Ben Dribus

Jonathan Cender

My appreciation for your repeated reading of my paper.

You ask some detailed questions. So folks don't have to go back and forth to the paper to refresh their memories, some basics. The Wallis zero, or "w" as you're calling it here, is an absence bar, "/", over the reciprocal of an array of the real numbers. The array is denoted by the infinity symbol since that is close to how it was first used by Wallis. So w = /1/(infinity).

"1. You say that "present arrays of the form n(infinity) only arise by division." I assume you can only mean division of the form n/w=n(infinity)?"

Correct, yes.

2. If this is true then (infinity) means 1/w?

Almost. It means 1/w = 1(infinity) since (infinity) is the unsigned basic array without any real term.

3. You distinguish "reciprocal" from "multiplicative inverse." Does this mean that the binary operation of division is not the inverse of multiplication in your "unreal number system?"

Unreal arithmetic is the same as real arithmetic except that division by zero is defined. Division in real arithmetic is a partial binary operation since division by 0 is undefined. Does this mean that division in unreal arithmetic is no longer partial? It will be interesting to see what consensus emerges. What seems relevant to me in answering this question is the following.

As you point out in your question, Ben, I distinguish between "reciprocal" and "multiplicative inverse."* I had to make a choice: keep the usual zero multiplication rule or break it and let zero have a multiplicative inverse. Well, if w has a multiplicative inverse, then the distributive property ends up not applying in important ways. Didn't want to lose that property! So even though the Wallis zero now has a reciprocal as in your question 2, zero still follows the usual rule that anything times zero is zero, and that means zero still doesn't have a multiplicative inverse.

So to try to sum up - perhaps we can say that unreal division is not only partially binary as with real arithmetic, but partially trinary as well.

*It may help to consider that neither 0 nor w have a unique multiplicative identity. Every real number, and still, every unreal number, is an identity element for zero.

Benjamin Dribus

Jonathan,

Thanks for the detailed responses. My interests in physics have led me through a lot of weird mathematical structures like algebras over sets with partially defined operations, etc., so I tend to take things like this seriously. This is the first potentially physically relevant structure I can remember seeing in which the distributive law fails. If I could find the time, I would like to try to get a more rigorous understanding of this. There may be similar structures lurking around unnoticed and waiting to be exploited. Take care,

Ben

Jonathan Cender

You're welcome. I hope the details helped.

Like you, I am unaware of algebras where the distributive law fails. The only fail now? [math] n\infty({\frac{1}{\infty} {\frac{1}{\infty})=n\infty({\frac{1}{\infty}) n\infty({\frac{1}{\infty}) [/math] This ends up as [math]n = 2n[/math]

While playing around with array numbers, other, more exotic, structures where the distributive law fails arose, but at the time I saw them as failures on the road to replacing 0 in real arithmetic. There may well be something of interest in terms of "non-distributive" partial algebras, however, and I may play around with them again.

Jonathan Cender

Even though it may be clear by now, for the record, here is my reply to Q4.

The infinity symbol denotes an array. So its reciprocal is an array number since an array is involved. And repeated division by zero does result in new numbers such as

[math] n\infty \div {\overline{{\frac{1}{\infty}}} = n\infty \bot^2 [/math]

The upside down "t" symbol [math] \bot [/math]

represents the degree of orthogonality to the basic "Penrose" array which is for all intents and purposes the real number line.

Thank you very much for your questions and comments and I look forward to reading about path algebras and their novel treatment of nothing in your essay.

Benjamin Dribus

Jonathan,

Thanks for the clarification. By the way, I don't really discuss these properties of path algebras in my essay, since I figured most of the audience wouldn't be thrilled with the details. But since you have already thought about these things, I thought I'd mention that they do indeed work that way, and that this provides another example of quite relevant weirdness involving 0 in physics. Take care,

Ben

Jonathan Cender

Interesting. I'll have to look into path algebras.

Steve Dufourny

Hello,

1/0 AND 1/infinity are intriguing. I beleive that that depends of utilized series. The zero is like the 1. Imagine the 1 like a connection with waves and so in a kind of dance of informative bosonic fields. The 0 like the nothing, so without an exchange of informative bosonic/fermionic dynamic. Turn off and turn on. Now if the finite groups are not utilized with the biggest determinism considering the serie universal of uniqueness(finite groups). So how the infinity, the infinities and the zero can be superimposed witha king of categorification. The volumes are relevant also like the rotations spinal and orbital. In deed the 0 and the 1 can be utilized like a pure harmonious oscillation. The series inside so become very relevant. It permits to correlate the serie of uniqueness and the volumes more the rotations. The informations of complementarity seems relevant.Now of course the puzzle, of synchronizations or sortings of evolution in a pure rational road of evolutive polarization m/hv, is very complex. The extrapolations of complexs dimensions(I have understood the fractal of 3D dimensions ! :))

The 1 and the 0 so can be taken like a pure gauge of distribution and taxonomy.

1/0 or 1/infinity need a real universal distibution.

The words of Riemann are relevant."Now it seems that the empirical notions on which the metrical determinations

of space are founded, the notion of a solid body and

of a ray of light, cease to be valid for the the infinitely small.Bernhard

Riemann"

Ther spaces appear in correlation with the universal entanglement of spheres. The 1 like primes. that said that does not mean that a mass possesses an infinite serie. On the opposite, they possesse a finite serie , so a real road of quantization proportional with rotating spheres and their specific number.

The 0 is not really a good partner in this line of reasong for the quantization of mass.

The nothing is not the 0, because the nothing does not exist. On the other side, it is good tool for the taxonomy of numbers and their spherical symmetries.

The serie of primes seems finite and not infinite. The 0 is not a prime.

Imagine 1/ serie finite of primes. and compare it with the irrational 1/0 and 1/infinity.

It is relevant when the finite groups are inserted like the classment. Now imagine a pure correlation with the volumes and the rotations, see that my equations E=m(c³o³s³) and mcosV=Const are relevant when the serie is finite about the fermions and bosons.

The 1and the 0 are tools , the physicality does not really insert the 0.Imagine the 0 multiplicate by the Universal sphere, I don't think that this universal will disappear. So it is an illusion, a simple tool.That said it is relevant considering the binar codes and the turn off, turn on. It is really a question of taxonomy of finite groups, infinities, infinity which is very improtant. If not we cannot quantize this mass inside a specific isotropical and homogene space time in evolution.

Good luck for this contes.

Regards

Steve

Steve Dufourny

Oh my God, you have chance to live there. If I could, I will put my mother there. Her health will be better :)

You know I find the 0 intriguing also. It is indeed relevant for the taxonomy of our numbers.

I ask me how is the distribution of primes inside this evolutive universal sphere.Considering the uniqueness of the serie and the spheres.

If we take a serie numerical, evolutive with a finite group. so we can see an oscillation of primes. The volumes seem essential.The 0 is not an universality in this distribution of the physicality correlated with these primes.I think that a maximum exists in the pure physicality and the number of spheres correlated for the uniqueness is finite.The volumes of the serie are very relevant.

The nothing does not exist in fact. I agree also about the words of Mr Dribus. That said the indetermination is so subtle when the infinity, the infinities, the constants,the finite groups are classed in a pure sphericality with a finite groups of primes. The volumes of the serie of uniqueness are very relevant in a pure spherical distribution of primes. The derivations seem relevant.

The 0 does not exist in fact for the physicality, that said,we use it in a pure symmetrical way.The nothing is not really physical.The numbers them are physical.is it a number so?, no at my humble opinion.It is just a tool helping for the taxonomy of numbers.

Regards

Sergey Fedosin

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