Is 1+1=2 an empirical proposition? by Yafet Erasmo Sanchez Sanchez

"Again, what gives life to the symbol game and eventually transforms it

into mathematics is the empirical regularity that must be associated with

the symbols."

That's cool.

What is the nature of the association?

Yafet, you begin with a musing on 1 1 = 2, the very definition of unassailable established truth in popular language, and how you would explain it to a child.

For a very young child you would assume this 1 and that 1 refer to distinct but interchangeable objects, so that the 2 in 1 1 = 2 refers simply to the counting number with which you would name the result of grouping them. But then you stumble across the problem of how to explain to the child as she grows older the deeper and more profound truth you see in 1 1 = 2 as a mathematical proposition.

I spent three years before the mast teaching algebra (and inevitably remedial arithmetic) to failing ninth graders, many of whom were very bright but angry working-class children destined for the scrap-heap of life. These youth had a deep hunger for a chance at life, not yet completely beaten, turning hopefully to me for help. My commitment was that I would find a way to teach them math that would make sense to them. Sadly, I found I could not. Time and time again I would lead a child who was stuck on something step by step forward from the last thing they understood (usually adding on their fingers.) Always we quickly came to some unprovable axiom which I was unable to persuade them to take on faith.

I searched the literature, and went off to three summer institutes on my own nickel, looking for someone who could show me the key. I found programs that worked based on building strong personal relationships (not an option in a modern public school environment), programs that assumed a basic understanding of arithmetic and the elements of algebra, and programs that tried to make learning fun, but nothing that met my need for a curriculum or "intervention" that would make sense. The best advice I got from the most successful teachers was to forget all that starry-eyed stuff and just make the kids buckle down and memorize and drill until they could do math without thinking!

The meaning even of 2 2 was a problem. Not when it occurred by itself, but when it occurred in some situation where the two 2's are used in different ways. Trying to understand it with the quantity model foundered on order of operations, where only good memorizing skills worked.

So my question to you is, why do you think there is a deeper meaning here? Why would you assume my ninth graders were wrong when they challenged me with "mister, that doesn't make any sense"?

Perhaps it doesn't make sense, except in very limited circumstances. But if it makes no sense as a general proposition, then the entire edifice of mathematics is a fatally flawed jury-rigged device, parts of which have been bent and beaten into a form that is with great effort useful to physics.

If this proposition is too horrifying for you to even think about, by all means ignore this comment. You've worked long and hard at mastering math as it is, a great achievement, and I don't want to ruin your life or your career.

But if you share a gnawing sense of something wrong and you long for a mathematics that actually makes sense, take a slow patient look at Rob MacDuff's deceptively simple submission "Mathematics of Science".

Is 1+1=2 an empirical proposition?

My response is: Yes of course it is. With this qualification: There is no such thing as a number without a thing. The thing is what is to be counted. The plus sign either means that countable objects are being counted or it is ambiguous and no longer tolerable as a mathematical symbol. Perhaps the number 1 and the number 2 in the title 'equation' are being misrepresented because they are written alone as if they were a mythical entity commonly referred to as a pure number of which I say there is no such thing.

Dear Yafet Erasmo Sanchez Sanchez,

I gave my opinion above and maybe it seems confrontational, but the fact is that I like your essay. It raises some points that I often see others debate. I don't see the debate as justified. Mathematics is not about chemical reactions any more than it is about physics. I see it as being about counting countable objects and providing shortcuts for counting. If a physicist wishes to mathematically describe the union of a photon and a particle of matter then they should count what is countable. As a matter of fact they do that: They count units of energy. I think that mathematicians and physicists who read it should debate your essay. I am neither but I will push your essay up so that it might gain notice. Your discussion about mathematics and String theory might do that here. It will probably be temporary and someone or someones will use some ones on it. In any case I can't rate it a second time so you can respond to me freely about your opinion should you choose to do so. I see no responses thus far from you and suggest that you might consider participating. Make your presence known by commenting on others' essays. In the end you should, of course, do what you think you should do. Good luck.

James Putnam

Dear Potential Dr. Sanchez,

I have no wish to be disrespectful to you or to your essay, but I think abstract mathematics and abstract string theory have nothing to do with how the real Universe is occurring for the following real reason:

Do let me know what you think about this: This is my single unified theorem of how the real Universe is occurring: Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of abstract NOTHING. Proof exists that every real astronomer looking through a real telescope has failed to notice that each of the real galaxies he has observed is unique as to its structure and its perceived distance from all other real galaxies. Each real star is unique as to its structure and its perceived distance apart from all other real stars. Every real scientist who has peered at real snowflakes through a real microscope has concluded that each real snowflake is unique as to its structure. Real structure is unique, once. Unique, once does not consist of abstract amounts of abstract quanta. Based on one's normal observation, one must conclude that all of the stars, all of the planets, all of the asteroids, all of the comets, all of the meteors, all of the specks of astral dust and all real objects have only one real thing in common. Each real object has a real material surface that seems to be attached to a material sub-surface. All surfaces, no matter the apparent degree of separation, must travel at the same constant speed. No matter in which direction one looks, one will only ever see a plethora of real surfaces and those surfaces must all be traveling at the same constant speed or else it would be physically impossible for one to observe them instantly and simultaneously. Real surfaces are easy to spot because they are well lighted. Real light does not travel far from its source as can be confirmed by looking at the real stars, or a real lightning bolt. Reflected light needs to adhere to a surface in order for it to be observed, which means that real light cannot have a surface of its own. Real light must be the only stationary substance in the real Universe. The stars remain in place due to astral radiation. The planets orbit because of atmospheric accumulation. There is no space.

Warm regards,

Joe Fisher

Dear Yafet,

Thank you for not reporting to Fqxi.org that my comment was inappropriate and by doing so have the Administrator classify it as Obnoxious Spam and remove it.

1. When I type the word "abstract" definition into the Google Search Engine, It gives a listing of definitions starting with: "existing in thought or as an idea but not having a physical or concrete existence."

Reality does not need an (abstract) ultimate judge of (abstract) empirical evidence. Following your reasoning are you (abstractly) able to (abstractly) model current (abstract) empirical evidence and (abstractly) predict new (abstract) empirical phenomena?"

I have a complete skin surface. You have a complete skin surface. Every real thing has a real complete surface of one sort or another. I know that no matter in which direction you look, you will only ever see a plethora of real surfaces.

Reality is. It does not need to be modeled, analyzed or scrutinized.

Joe Fisher

Yafet Erasmo Sanchez Sanchez,

"...how does the zero number fit with the idea of number if there is nothing to count?"

I can't think of any case where zero is a number. If I am to make sense of zero it is not as a number but as a word indicating that I have not yet begun to count anything. If zero appears on a line with positive numbers to the right and negative numbers to the left, it remains a point of origin where I have not yet begun to count. If I count three units to the right and then reverse my direction and count five units to the left, I pass zero but ignore it in favor of the number three. In other words, zero doesn't exist as a part of the act of counting. The useful function of zero is as a placeholder so that I do not need to be personally located at a particular place in order to start counting from that location. The zero sometimesrepresents me standing stationary or is my marker for where I am imagining I am standing stationary. I could of course be speaking on behalf of another object other than myself.

"Also, do you have any insights about ordinal arithmetic? There certainly we are talking about "counting" in some sense. However, it is infinite counting which I am not sure is easy to represent with physical things. What are we counting then?"

No I don't, but, I do know that we receive all information from a storm of photons impacting upon us. Those photons along with all the other photons in the universe add up to a countable number. Furthermore, each photon delivers an incremental measure of change of velocity of a particle of matter. The photons are tiny bits of information. Out of that wild mix of photons arriving from innumerable sources, we know by innate means how to discern patterns of importance to us. Those patterns form in our minds. While they are there, we can imagine that they are continuous and perhaps even infinite, but, in no case did those photons originate from a source that is either continuous or infinite. My point is that there is much math that is useful to apply to the patterns that we imagine in our minds and then there is the math that applies to the source of our original information. I think that those maths are not the same maths.

"...You encourage me to read your essay... ."

That would be great, but, it might appear to be one very wild ride. :) I can't be right unless theoretical physics is wrong beginning with its treatment of mass in f=ma. After I eliminate the indefinable status of mass, there are great changes in store. I have publicly written about and presented many of those changes. I think I am on the correct path, but, let me know what you think. Thank you.

James Putnam

You have titled perfectly, also made some honorary effort of gatherings.

-With regards,

Miss. Sujatha Jagannathan

Dear Yafet,

Your short and well structured essay is interesting. I agree that the concept of meaning is very relevant in this discussion about the comparison of maths and physics. Let me know if I understand you well that meaning arises in a context. You can find in my essay, p.6, a Frege's quote "Never ask for the meaning of a word in isolation, but only in the context of a sentence". Contextuality is a very important concept in QM.

You talk about a multiplicity of meanings of string theory (ST), this often taken as a weakness of ST but it seems that your opinion is different. I wonder if it is not precisely the multiplicity of meanings of QM that makes this theory so rich. The philosopher Popper considers QM as non falsifiable may be as ST.

I hope you will have time to go through my 'moonshine' topic.

Best,

Michel

Yafet,

Are you familiar with the Wittgenstein-Turing colloquy at Cambridge in the late 30s re: building a bridge so it won't fall down? Cuts to the core of the whole mathematics vis-a-vis physics debate.

Dear Yafet,

Thank you for a short, to the point and well argued essay: I have read more than half the essays in this contest, and yours is one of my favorites. As a fan of Wittgenstein, I totally agree with you when you say:

"The main shift one would like to achieve is to move from the dichotomy of "true" and "false" propositions to the notions of 'sense' and 'nonsense'."

I quite liked your analysis of the question "What kind of endeavor is string theory?" in the closing paragraphs of your essay.

I find it strange that your essay has attracted so few votes so far. I hope that the rating I will give it will move it higher in the community rankings, and will give it a better chance to get noticed.

All the best,

Marc

P.S. My essay tackles the philosophical question of the ultimate relationship between "All of mathematics" and "All of physical existence": I hope it makes more "sense" than "nonsense"... ;)

Hi Yafet,

I think understanding what mathematics really is all about will definitely inform physicists, so I thank you for you thought-provoking essay. Thoughts about Euclidean geometry come to mind as an area of Mathematics that you might have considered to be "hardened" at one point... that is, until Non-Euclidean geometry came along. Even the terms "point" and "line" can take on different meanings if one imagines different non-Euclidean spaces. Should physicists not be concerned about philosophical discussions around the meaning of symbols they write down, so long as most of their peers can extract the essence of what their formulas and definitions are suggesting?

Please check out my Digital Physics movie essay if you get the chance. There are some questions posed at the end of the essay that may interest you. Here is a couple that came to mind after reading your essay.

12) If I created a very simple formal system in which almost every statement was undecidable, and then I took advantage of this fact by choosing some of the most counterintuitive independent statements to add as axioms, is there any value in physicists studying this area of mathematics?

13) On the other hand, how can some statements in a formal system be considered more intuitive or self-evident than others if any string of symbols should be looked at as being devoid of meaning?

Thanks,

Jon