Thought Experiments in the Abstract Field of the Mathematics of Infinities Produce Experimental Artifacts Suggesting That Their Use in the Real-World Science of Physics Should Be Reexamined by Roger

Roger Granet

Essay Abstract

Here, I question the assumption in set theory that the size of an infinite set is the same as the size of an infinite subset derived from it as well as the validity of using this result in physics. Consider a thought experiment in which one wants to study the properties of the single set of the positive integers within the framework of that single set system, and one wants to compare the total number of positive integers in the set to the total number of even integers. The traditional experimental processing method extracts the even integers, puts them into a separate subset and pairs off the subset's and set's members one-to-one with a function. After finding no elements left over, the original set and the subset extracted from it are said to be of the same size. However, this method dramatically alters the original single set system and ignores the inherent relationships of that system in which every even integer is always accompanied by an adjacent odd integer. When one takes this relationship into account, one finds that there are twice as many positive integers as even integers within the single set system. The traditional processing method produces an experimental artifact. This should be unacceptable in a well done experiment. While this may be acceptable in the abstract field of mathematics, one cannot simply study a subset of a real, physical system in isolation without considering the inherent relationships between the subset and the rest of the system. And yet, the mathematics of infinities is based on doing just that and is used extensively in physics. I suggest that physics should require the use of聽 proper experimental technique, especially in its logical foundations.

Author Bio

I have Masters degrees in biochemistry and business and work in Columbus, OH (USA) as a biochemist. I have a lifelong interest in thinking about the question "Why is there something rather than nothing?". Set theory, including topics related to null and infinite sets, are related to this topic, and this essay sprang from thinking in those areas and from my background in biochemistry. Other essays on this question and infinite sets are at: sites.google.com/site/ralphthewebsite

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[deleted]

Dear Roger Granet,

Having thoroughly dealt with Cantor's naive set theory, I feel confirmed by Ebbinghaus in that Cantor was simply wrong. Since you are also questioning set theory, I would appreciate if you were ready to comment on my views. Did you get aware of other allies here?

Regards,

Eckard

[deleted]

Eckard,

Hi. I just returned today from a several-day long vacation so that's why I haven't responded sooner. I don't remember seeing any essays this year that have really done much with set theory as it relates to physics, and in terms of allies, you're the only that's posted any kind of a reply to my essay at all. Even among "crackpots", I seem to be a crackpot! :-) But, I have seen several essays that have questioned the seeming supremacy among physicists of mathematical rather than physical/mechanical thought. That seems to be kind of a general theme among many of the essays. To me, I'm a little disappointed in the reasoning ability of physicists (at least the ones I can understand) and mathematicians in that they:

o Make a lot of unfounded assumptions.

o Many seem to claim that mathematical constructs or physical laws exist "out there" somewhere and can serve as the first cause of the creation of the universe. Logically, this is no different than assuming a "god" created the universe. It's possible, but they need some evidence other than their say so.

o They use mathematics as the basis of physics and yet use very poor experimental technique (in my essay) in their mathematical thought experiments.

In trying to make some progress in basic questions, I think we should try to avoid all, or as many as possible, assumptions and start our reasoning from basic logic. While I may not achieve it, that's the approach I've tried to take with my thinking.

What was the name of your essay? I'll go back and read it more carefully. I had scanned them all up through about 4-5 days ago, but hadn't read too many thoroughly due to work and family time constraints.

Thanks!

[deleted]

Roger,

You didn't yet comment on my [link:fqxi.org/community/forum/topic/1364] views[/link).

What about the property of an infinite quantity to be not larger than an also infinite part of it, you will perhaps agree on oo 1 = oo. Incidentally, Georg Cantor was not the first one who used the self-contradictory expression "infinite number". Even Weierstrass used it.

Eckard

[deleted]

Hi Roger,

I just read your essay, and while I am not a mathematician (I do study physics) your argument totally makes sense to me.

Have you considered discussing this with some mathematicians? If you have but your argument was dismissed without a solid reason (e.g. proof) for why it has no merit, I would go to another mathematician.

Unfortunately my knowledge in this area is very little, but if your argument is repeatedly dismissed by mathematicians without solid reason, then I suspect that you may have discovered a novel distinction that is not yet part of our established mathematical knowledge.

In that case, I would totally change the perspective, or punchline of your argument. Instead of framing it as a criticism of sloppy mathematical thought experimentation, I would frame it as the discovery of a novel distinction between infinities, namely those, coming from a common set that has globally fixed relations between its members, and for which, for instance, ratios would be meaningful, and others for which such relations are undefined.

If you can derive interesting mathematics from this new distinction, then it is more likely that it will become part of established mathematical knowledge, and if it ever does become part of established mathematical knowledge, your original argument will be that much more forceful.

Unlike in physics, in mathematics you make up the ground rules from which to derive new results and prove new theorems , and I believe one of the best ways to do this is to introduce new distinctions. This reminds me of a brand new announcement of the solution a long-standing very difficult conjecture by a mathematician, who, in order to solve it, created in effect his own mathematical universe. You can read more about here:

http://www.math.columbia.edu/~woit/wordpress/?p=5104

The only criticism I have (and it has no bearing on the merits of your argument) is your highly unorthodox usage of the term "reference Frame". I would encourage you to consider using different terminology as this may put off people and deter them from considering your argument further.

To me, a more appropriate term would be "reference standard" (typically we would consider a ruler or a clock as a reference standard, and that aspect of the observer which allows him to obtain information about the world a reference frame), but other terms might serve as well.

It is probably very rare that a non-mathematician discovers a novel mathematical distinction, especially if it has implications for physics, and I find it very interesting that, finding myself in this regard in a similar situation as you, I came across your work. If you are interested, you can watch a video where I introduce a simple mathematical distinction which is as yet currently unrecognized (to my best knowledge) in the first 7 minutes or so of a talk I gave at a quantum physics conference. I believe that recognizing the distinction has profound importance for understanding the meaning behind the mathematical formalism of quantum mechanics.

http://youtu.be/GurBISsM308

My essay also contains an illustration of this distinction in a more accessible way.

I'm glad I came across your paper, it is unfortunate that your argument is not widely appreciated (I know exactly how that feels), I sincerely hope that this will change.

All the best,

Armin

Roger Granet

Armin,

Hi. Thank you for the feedback! I've tried many times to get mathematicians, physicists and philosophers to listen to this and other ideas with almost no success. I think very few even read or seriously think about the points raised and just object because I'm an amateur and the idea goes against accepted thought. Most of the few comments I have received reflect very poor reasoning (infinities are just different; thought experiments don't need to follow proper experimental methods, etc.) and give no evidence for their arguments. The one positive comment I had was from an academic mathematician working in experimental mathematics in Canada.

Your point about rewording the essay to be less a criticism of the use of experimental technique in mathematics than about suggesting a new distinction between different kinds of infinities makes a lot of sense. I will admit that whenever someone starts an argument with a criticism, it's psychologically harder to listen to the merits of his/her argument. So, maybe some marketing psychology would help! The use of the phrase "reference standard" is also a good one. I'm kind of used to thinking of what reference frame/perspective something is being observed from that maybe that's why I used the phrase "reference frame".

I may use your ideas and rewrite the essay, but my main goal is to work on the question of "Why is there something rather than nothing?" and to use anything I get from that in building a working model of the universe that might be able to make predictions someday. I first approached this area by trying to understand the null set, infinitely small things and infinitely large things. That led to not only the current FQXi essay but the last one, too.

I read your FQXi essay thoroughly and found it very good. I'll make my comments over at your essay but will say here that I think your way of thinking where you try to visualize what things might visually look like to observers from different perspectives (what I've called "reference frames" in another essay), like from a two-dimensional or a three-dimensional perspective, is a very good way of thinking. Because the minds of mathematicians and physicists (and everybody) are observing things from an existent, finite (not infinite), and 3-dimensional perspective, they can easily miss things that might not be existent, finite or three-dimensional. I've tried to do something similar in the last essay contest where I talk about how an infinite set of finite-sized balls spreading out in all directions might appear to a finite observer within the set and to a hypothetical, infinite observer outside the set. The different views of this set as discrete and continuous, respectively, may have relevance to our different descriptions of our universe as discrete or continuous?

By the way, I'm from Michigan originally, too, and have a friend that lives in Ann Arbor.

Thanks again for the feedback. As you mention, being ignored by academics can get very frustrating!

[deleted]

Roger

have you read my essay?

http://fqxi.org/community/forum/topic/1413

[deleted]

Hi. Thanks for the comments. I scanned over your essay, and while I'm not sure I understood it because of language issues, certain quotes like the following:

"It would be more reasonable if we develop an essay contest to contest this determination on the basic theory with specific questions, eg: a specific definition and detail to the universe? true nature of gravity? or principles form a habitat in the universe? with the requirements and clearly defined assumptions with each individual involved."

gave me the feeling that the main point of your is that we really should try to define and figure out the most basic things before we move on to making speculative new theories in physics and philosophy. If that's what you were trying to get at, I totally agree. For instance,

1. I think one of the problems with modern thought on foundational questions in physics and philosophy is that we need to remember that we're trying to describe the physical, 3-dimensional universe. That means that all our basic equations and postulates should be describing some physical 3-dimensional stuff. That is, if we postulate that there is some fundamental particle or force, it needs to be a physical, 3-dimensional particle or a force carried by 3-dimensional particles of "stuff".

2. Speaking of force, what is force? If the universe is made of physical, 3-D components, force must be some change in these units. The most basic change I can think of for a 3-D thing is causing it to change its shape. So, IMHO, force is some mechanism that can cause a change in shape in a 3-D component of the universe.

A. For the electromagnetic force, what is the physical mechanism for how two like charged things repel and two oppositely charged things attract? How does the exchange of photons or other "force particles" physically, 3-dimensionally cause this? If I throw a ball back and forth to someone, that doesn't cause me to move towards or away from them.

These are just examples, but this type of first principles, physical/3D-type thinking is what I'd like to see more of in all of physics and philosophy. I try to use it in my thinking as do some of the other essay writers. Let's spread the word!

Benjamin Dribus

Dear Roger,

I think you are pretty much right on target (and I'm a mathematician!). The great British physicist and mathematician Roger Penrose has asked a lot of similar questions and raised a lot of similar points, although I haven't read most of his work on these subjects first-hand. In particular, there are a lot of different ways of measuring the size of sets, and people often focus only on the "cardinality," which is in some ways the coarsest way of measuring and tells you the least information. In terms of cardinality, it's true that the even numbers are the same size as the integers, but as you point out, the whole point of identifying this set as "even integers" specifies the set of integers itself as the "parent set," as you put it, which means you are really referring to two sets rather than one. I would have to agree that it's rather idiotic to then proceed to ignore the relationship of the set to its parent set when talking about size.

My own attempts at fundamental physics involve mostly countable sets (sometimes finite) and issues like these arise all the time. In particular, I often have to think about one "partially ordered set" as a subset of another. Often these sets have the same cardinality, but are definitely not the "same size." If you're interested, you might look at my essay On the Foundational Assumptions of Modern Physics. What I try to do is build up fundamental physics from simple principles like cause and effect. This leads to some rather thorny mathematics, but my view is that the physical principles ought to be simple and well-motivated, and the math ought to be whatever it has to be to get the job done.

Anyway, I enjoyed reading it! Take care,

Ben Dribus

[deleted]

Ben,

Thank you for the nice comments! You should be careful, though, saying nice things about a "crackpot"! They might not let you finish the PhD! :-)

I think your way of thinking shown by your saying "What I try to do is build up fundamental physics from simple principles like cause and effect. This leads to some rather thorny mathematics, but my view is that the physical principles ought to be simple and well-motivated" is exactly right and I wish more physicists and thinkers in general would think this way. If we start at the base principles like cause and effect, we have a better chance of building a working model of the universe that can make predictions than by starting out with high level, assumption-riddled, current physics thinking and working down to more fundamental levels.

I guess one of the main points of my essay is that it's real important to consider the "physical" relationships between elements of a system when analyzing that system. These could be relationships between elements within the system or between the elements of the system and the "observer" (ie, the mind of the mathematician or the person doing the physics experiment). For the mathematics that describe physical systems, the mathematical objects themselves should have some "physical"-like attributes so that they can embody these relationships between the elements. From reading your essay, even though it was way over my head, it sounds like a relationship between two elements in a set like what I suggested with the even integer and its accompanying odd is kind of similar to your binary relation? And, if one element causes the related element to appear, is this a causal relation? If this understanding is right, this makes a lot of sense to me because my own view of existence is that given a fundamental unit of existence, whatever that is, this unit will cause the formation of identical units around it, these new units will cause the formation of new units around them, etc. and this expanding space of units is equivalent to our universe. So, in my view, I think I would say that there's a causal relation between each unit of existence and the units it causes to appear next to it. I have more on this at my website at

https://sites.google.com/site/ralphthewebsite/filecabinet/why-things-exist-something-nothing

Back to your essay, while these comments may be totally off-base because I don't have the math background to understand much of it, I would say that:

1. I totally agree that the assumption that systems evolve with respect to an independent time parameter, and that the universe has a static background structure seem unlikely. To me, time is just the same as a sequential chain of physical events with the earlier events corresponding to earlier times. If time exists as this separate dimension somewhere, I'd like someone to point it out to me now. Where is it?! Also, in regard to the second assumption, I think of the universe a little more holistically where matter and energy aren't occurring against a separate space background, but rather that they're interactions between the units that make up the universe and space.

2. You mentioned on pg. 6 "This means, in particular, that spacelike sections are merely unordered sets, with no independent notion of distance or locality". If I understood this, I think I'd also agree because I think that location of something refers to its position relative to other things within a bigger set of things. That is, while a single existent state may "be" a location, it doesn't "have" a location within a bigger reference frame.

3. My own view on volume is that to physically exist, any existent state must have three dimensions, and, therefore, volume. I have trouble imagining an actual physical state in which one of the dimensions is zero. Not just infinitesimally small but actually zero. At zero, it disappears. So, three dimensions, or volume, seems to me to be a requirement of an existent state and thus a requirement for whatever existent state makes up our universe.

Sorry for the long response. Thanks again for reading my essay and the comments!

Roger

Benjamin Dribus

Dear Roger,

Thanks for the kind remarks... I also posted a similar reply to your comments over on my thread, but thought it would be easier if they were here as well. I took a look at your website, but unfortunately only the top of the page would load; I don't know if this is a site issue or a browser issue.

I think your point about considering the "physical" relationships between elements of a system is of crucial importance when one is trying to do physics. In a way this should be obvious, but the "mathematization" of physics has led many people to ignore the fact that physical ideas or principles are something fundamentally different. Math has no notion of "cause and effect," for instance, even though you can model the relationships between causes and effects mathematically.

I appreciate your remarks on my essay. Even if some of the mathematical content was a bit unfamiliar, it seems that you understand quite well what I am trying to do conceptually. In particular, I'm trying to give a precise description of something very like what you mentioned in your point 5, with the clarification that I think each "element" generally has multiple "parents." In particular, by "causal relation," I mean almost exactly what you said.

One difference we might have is that I think dimension (like "space" and "time" themselves) is just a "way of talking about what actually happens." For instance, in three-dimensional space you can "go in six different directions," forward, backward, up, down, left, right. If you turn this around and start with a bunch of events that are related to each other in this way (each having "six neighbors" in an obvious sense), then you would get a "three-dimensional network." This is all a very rough and imprecise way of describing things, but hopefully gives the right picture. I think that the dimensionality of the universe is telling us something about how interconnected the structure is at the fundamental scale: how many "direct neighbors" each "fundamental element" has, and how they are arranged. All this ignores the quantum-theoretic version, of course.

Take care,

Ben

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