Christopher,

Consider the following question: Why Superman doesn't need a boss?

Because he has super vision.

I think mathematics is a language similar to English or any other natural language in many aspects. Now, if you allow me to extend the analogy as far as I like I think the problem your students are having is similar to the problem some students learning English have when they encounter homophones or to understand the double meaning that is common in jokes.

Of course someone might think as nonsense the claim " I smell a rose." if it is unaware that there is a flower with that name and only is used to hear that word in the context of something moving upwardly in the past (the past of rise). Failing to understand the multiplicity of meanings doesn't allow to make sense.

I don't think your students are wrong but that they are similar to a kid that demand an explanation of how come the same string of letters have different meanings or someone asking us to explain a joke. As you pointed out ":"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."

This different meaning of the word are related to different contexts and uses of the number two. As a teacher you are aware the delicate and difficult task to convey this to your students. I am in no way an expert and probably only pedagogical research can help you to learn new techniques how to be more effective to show the different contexts in where different uses of mathematical symbols are used. Saying that I would be happy to hear about the specific situation your ninth grades regard as nonsense.

Regarding doing math by memorizing without thinking I would say I have a divided opinion. I think is not healthy if it extinguishes curiosity for more complex subjects or problems. But at the same time knowing thing by heart are useful and not necessarily bad if we are aware of the efficiency they have in our life. Perhaps a ninth grade wont be able to give a proof of why she should received $1 dollar change for a $9 dollar toy if she is paying with a $10 dollar bill. But, certainly, knowing that would be very useful, even if she only knows it by heart.

I have read the essay you suggested and I would leave a comment there when I have more time.

Finally, don't worry about ruining my life or career. Discussion is important and I always welcome it. As Wittgenstein said "A philosopher who is not taking part in discussions is like a boxer who never goes into the ring."

Dear Joe,

I am a little confused by your arguments. How are you defying abstract? Also, as much as I like or dislike your arguments the ultimate judge is empirical evidence. Following your reasoning are you able to model current empirical evidence and predict new empirical phenomena?

Kind Regards,

Yafet

Dear James A Putman,

Thank you for your comment. I agree number are used for counting, but how does the zero number fit with the idea of number if there is nothing to count?

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?

You encourage me to read your essay and I will leave a comment there as soon as I grasp the ideas you are presenting better.

Yafet

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

6 days later

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

          Hi Jon,

          I think the point you raised about geometry is very important; unfortunately I am afraid I have not a complete clear view of how to tackle the whole geometry in terms of use and meaning. Nevertheless, I would like to share some ideas I have about it.

          I think I would still say that Euclidean geometry is completely harden.

          The existence of other geometries didn't prove that the propositions of Euclidean geometry are false. At least not in the same way we say that the proposition 'the Earth is flat' is false. Rather we become aware of the background needed to use and therefore have meaningful mathematical propositions in Euclidean geometry. The background needed is the context of 'flat Riemannian space' (and all the contexts associated to it).

          As you mention the meaning of point and line takes different meaning in different non-Euclidean spaces. Seeing the differences and similarities in use of those concepts in geometry is becoming aware of the backgrounds needed to develop useful propositions in geometry.

          I also think that physicists should be concerned about philosophical discussions around the meaning of symbols they write down. I am sorry if my essay give you the wrong impression. The more physicist are aware of the meaning of the symbols they write the more they will be aware of the background and contexts where physics make sense. This clarity of background will allow us to identify useful empirical statements. This is not related with the true or falseness of the statements but rather with the appropriate context. As an example: The statement 'Romeo loves Juliet' is meaningless (it doesn't even make sense to ask if it is true or false) in a physics context, it belong to a literary context. We recognized the difference of background. However, there are more problematic statement that only philosophical discussion can clarify.

          I will read your essay and comment on it. It looks interesting. Regarding the questions you pose. I think for example question 12) what is asking is to find a use in physics. That means until we can argue that your formal system with the extra axioms can be linked to the context of experiments and predictions there is not a priori physical meaning. Question 13) is asking for why does human regard some thing more intuitive that others. I don't have an answer. I believe that if what we want is an explanation then advancements in psychology, biology, sociology, neurobiology can provide answers. Nevertheless, at the level of description we are aware of this fact. This is a fact about human nature. If humans would have different intuitions our forms of live will be different.

          Certainly, I will keep thinking about your post and if something becomes clearer for me I will let you know.

          Kind Regards,

          Yafet

          Dear Michel,

          I was not aware of Frege's quote. I liked it. I am not sure if I agree with Popper. Isn't the correct prediction of say the electromagnetic fine structure constant a sign of falsifiability. Or am I missing something?

          Also I think it is a great philosophical exercise to think about QM and the context needed to make it a physical science.

          I will read your 'moonshine' and comment when I finished it.

          Kind Regards,

          Yafet

          Dear Mark,

          Thank you for your comment. Yes, Wittgenstein rocks.:) I like you enjoy my essay and even more if copying Wittgenstein: 'I was able to not spare you the trouble of thinking, but instead encourage you to have independent thoughts.'

          I will read your essay and comment later.

          Thank you for your support.

          Yafet

          You are doing very well in this contest, but, you need 3 more ratings at the least. Hopefully, after receiving them, you will remain in the top 30. If not then you will need to work to get more ratings. I think that your essay can continue to rate high enough to enter into the finals.

          James Putnam

          7 days later

          Dear Yafet,

          Yours is a very subtle essay that gets to one of my favorite philosophical themes: Meaning.

          I have noticed that many times discussions fail to get to satisfactory conclusions because the participants do not seem to recognize that while they are using the same words, they attach different meanings to them, and a particular example comes up right in this contest: Your discussion of the meaning of the proposition 1+1=2 is exactly apropos to Edwin Klingman's (and his proponents) claim that he "debunked" Bell's theorem, which is at best (i.e. if correct) a more complicated version of your example of the two drops merging into one not being an applicable real-world example of the mathematics.

          I am surprised by the low response rate to your essay. If I may speculate, perhaps the title of your essay turned off some people because to most people the answer to the question it poses seemed so obvious that they didn't bother to see your argument that it really isn't if one is completely divested from the world we live in (again your merging-drops world is a great example). The subtlety of your argument may have escaped some, or perhaps your subtle criticism of ordinal arithmetic and string theory at the end might have antagonized others.

          Be that as it may, I found your essay a very thoughtful exploration of meaning in relation to mathematics and physics.

          Best wishes,

          Armin

            Yafet,

            Really nice essay. I think it is important and most difficult to understand the relation of physics and mathematics by simple examples. I basically agree with all your are saying in your excellent essay.

            One thing you left out in your analysis is the Kantian question: "What did we already accept in order to think that mathematics or physics is possible?"

            You say that: "The truth of a mathematical propositions is independent of any physical phenomena." I would say, that in order to be able to do mathematics some physical conditions have to be met: constancy of certain phenomena. Maybe also the structure of time.

            In physics the situation similar, when we say, that physics is an empirical science. The structure of time seems to be a precondition of experience.

            Maybe also the conservation of energy (Time translation invariance): Once Heisenberg heard about an experiment that showed, that the energy was not conserved. He reacted instantaneously: "Impossible!" How could he know that? How comes that some physical (empirical) statements although empirical seem always to hold true in our experience? The Kantian response to that is: because they are preconditions of objective experience to be possible.

            Again I think you wrote a great essay and I would love you could read mine and comment on it.

            Best regards,

            Luca

              6 days later

              Dear Akinbo,

              Thank you for your post. I like your example of the magician. But I think we should be careful between mathematical statements and empirical statements. Let´s say I answer two and the magician show me there are none. I will not be tempted to say mathematics are wrong, but rather that I am applying the wrong rule.

              If think behave in the strange way you propose certainly our arithmetic would be very different. I will coment on your essay. It looks very interesting.

              Kind Regards,

              Yafet