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

        Dear Armin,

        I appreciate your interest in my essay. There might be many reason why I have such a low response. Anyhow, I enjoyed thinking about this topics while doing the essay and I certainly will continue.

        Kind Regards,

        Yafet

        Dear Luca,

        Thanks for your comment. Kant is one of my favorites. I agree with you that to do mathematics we need a background or context and that is provided by the regularity of nature and our psychology. Although the true of such statements is not. But notice, that without the background, the true become irrelevant or nonsensical.

        I think Heisenberg, didn´t know. He believed so. But remember, how many things we consider impossible until they actually happened. Nevertheless, I believe you are asking the deeper question, how can we actually come to realise such regularities. I am afraid I don´t have an answer for that.

        I will read your essay and comment on it. Meanwhile enjoy this video from the birthday boy.

        https://www.youtube.com/watch?v=rzpL_5CI0WQ

        Kind Regards,

        Yafet

        Dear James,

        Your essay certainly is a wild ride. I think the effort you put to understand mathematics and nature from another point of view is remarkable.

        If I understood correctly you want to make physics concepts as empirical as possible and you claim that:" The general theory of relativity presents space-time to us as a real property.

        There is no empirical evidence to substantiate this claim."

        If the detection of gravitational waves becomes a fact how would that change anything of your discussion?

        Kind Regards,

        Yafet

        Dear Yafet,

        It appears that my comment and your response disappeared before I had a chance to see what your response was (except for the beginning, which can be seen in the side bar). I wonder whether the moderator can reinstate them?

        Armin

        Hi Yafet,

        your reply on my comment also got lost before I had the chance to watch the video you posted. Maybe you could repost it.

        You are right, Heisenberg believed only that energy conservation holds. But it is as right as we cannot have absolute certainity of anything. But also we cannot doubt everything. So if we believe physics is possible, we might believe that the laws of physics are time translation invariant and from that energy will be conserved.

        However you might like the film "The Oxford Murders". A film about a serial killing involving a Wittgenstein expert. Where the induction problem plays an eminent role.

        Best

        Luca

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