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FQXi FORUM

March 17, 2018

CATEGORY:
Questioning the Foundations Essay Contest (2012)
[back]

TOPIC: On the Hypotheses of Nothing Which Lie at the Bases of Mathematics and Their Consequences for the Foundations of Physics by Jonathan Cender [refresh]

TOPIC: On the Hypotheses of Nothing Which Lie at the Bases of Mathematics and Their Consequences for the Foundations of Physics by Jonathan Cender [refresh]

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.

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.

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.

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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/

L

ibertarian 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.

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/

L

ibertarian 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.

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 F_{1} is Newtonian field strength, then F_{1}' is gravitational force. I state that we have probably been mistaking F_{1}' for (C-F_{2})' so that the Einstein field equation with the cosmological constant should read similarly, i.e.

where the constant term is equated to a potential for energy of the vacuum and L_{uv} 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 G_{uv} 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

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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 F

where the constant term is equated to a potential for energy of the vacuum and L

Where I see this relating to your concept is that while a stress energy tensor for G

Regards,

Jeff Baugher

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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.

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

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

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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

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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...

view entire post

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...

view entire post

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

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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

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You're welcome. I hope the details helped.

Like you, I am unaware of algebras where the distributive law fails. The only fail now? This ends up as

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.

Like you, I am unaware of algebras where the distributive law fails. The only fail now? This ends up as

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.

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...

view entire post

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...

view entire post

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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,...

view entire post

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,...

view entire post

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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 and was the quantity of people which gave you ratings. Then you have of points. After it anyone give you of points so you have of points and is the common quantity of the people which gave you ratings. At the same time you will have of points. From here, if you want to be R2 > R1 there must be: or or In other words if you want to increase rating of anyone you must give him more points then the participant`s rating 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

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Sergey Fedosin

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