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A question from a rookie on MO.

During my research, I introduced in my physico-mathematical equations some "nasty", "problematic" mathematical objects (namely functional integrals) that are at least formally/symbolically defined but way beyond my level of understanding.

  • Perhaps they are already (well) known in theory $X$ as object $Y$;
  • Perhaps they are unidentified mathematical objects up to now;
  • Perhaps they do not make sense;

This I would like to know.

Those objects arise in a seemingly violent collision between dynamical system theory and Bayesian probability theory. Please have a look to question:

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)

By definition, if I were able to (well) define them, there would be no question at all.

Since my objects are functional integrals, you can think for instance about Feynman's path integrals to illustrate the situation: brand new mathematical objects only formally/symbolically defined at the beginning, takes time to make them well-defined.

So I've submitted one more question on MO, namely

https://mathoverflow.net/questions/237084/sums-series-integrals-whats-next

where:

  • I explicitly ask about the definition, the existence or the non-existence of my formally/symbolically defined hypothetical mathematical objects in some theory I unfortunately do not know;

  • I provide links to other MO questions where those hypothetical mathematical objects naturally arise, in order to show they do not come from out of space;

But I finally only get comments like:

  • Is it well-defined? No. Why not? Because you haven't defined it --- nor even given a hint as to what you mean or what properties such an integral should have.

  • *What properties do you want this operation to have? If you have a set of desiderata, they should be included in the question. If you don't, then of course the answer is that it's easy to define the operation any way you like (might as well take it to be identically zero, for example), and it's also easy to see that this question doesn't belong here

What if Feynman would ask about his freshly introduced path integral on MO today?

Is it well defined? No.

Why not? Because you haven't defined it

Yes I know, but at least it is symbolically defined!

How to escape this vicious circle? At which point Feynman's path integral was sufficiently well/ill defined to deserve consideration from mathematicians?

So please, how should I reformulate my question in order to get answers like "your object are in fact already known as $Y$ in theory $X$" instead?

FYI, I've asked the same question on SE:

https://math.stackexchange.com/questions/1753921/we-have-sums-series-and-integrals-whats-next

and here I get more constructive answers like:

Possibly functional integrals or (in QM) path integrals

It should be the converse, I guess. What's wrong?

Thanks for your help.

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    $\begingroup$ The comments gave you very good constructive advice. Listen to them instead of dismissing them. $\endgroup$ – Emil Jeřábek Apr 23 '16 at 8:34
  • $\begingroup$ @Emil. Be sure I'd like to do! But I am simply not in a position to do much more without any input, hence the question! $\endgroup$ – Fabrice Pautot Apr 23 '16 at 8:44
  • $\begingroup$ @Emil. How to escape this vicious circle? That is the question. $\endgroup$ – Fabrice Pautot Apr 23 '16 at 8:47
  • $\begingroup$ @Emil: as I said, what would happen if Feynman would ask today about the existence of his path integral on MO? Is it well defined? No. Why not? Because you haven't defined it. Yes I know, but my functional integral is at least formally defined! How to escape this vicious circle? At which point Feynman path integral was sufficiently well/ill defined to deserve consideration from mathematicians? $\endgroup$ – Fabrice Pautot Apr 23 '16 at 8:53
  • $\begingroup$ @Emil: Feynman's example added in the body. $\endgroup$ – Fabrice Pautot Apr 23 '16 at 9:09
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    $\begingroup$ Wouldn't Feynman have been responsive to the comments you got? Giving a hint as to what he had in mind, perhaps along the lines described here: en.wikipedia.org/wiki/… -- I don't think he would have left it at "but it's formally defined!" $\endgroup$ – Todd Trimble Apr 23 '16 at 12:25
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    $\begingroup$ It seems to me that you are not using the phrase "formally defined" in a way that I think most mathematicians would use it. For most mathematicians, for something to be be formally defined implies at least that it is defined, and probably defined with respect to some clear axiomatic foundation, hence "formally" defined; but in your case, the basic notion does not yet seem to be defined in any way, either formally or informally. What you have is a vague idea or a metaphor rather than a formal definition. Metaphors and ideas are often valuable, but they are not the same as formal definitions. $\endgroup$ – Joel David Hamkins Apr 24 '16 at 1:56
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    $\begingroup$ @Joel The word "formal" appears to be used in mathematics with two, almost opposite, meanings. One is the one you explained. The other one is that "formal X" is a symbolic manipulation that superficially has the form of an honest X, but is not actually valid (because prerequisites are not met, or we work with a different kind of objects than we should, etc.). Having said that, the OP is nowhere close to what most mathematicians would consider a "formal definition" even in the second sense. $\endgroup$ – Emil Jeřábek Apr 24 '16 at 9:07
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    $\begingroup$ The question is also formulated in a ridiculously general way that does not seem to have anything to do with what the OP really wants. That includes all the $\beth_i$ nonsense. There are just way too many ways of "summing $\beth_2$ objects". For example, consider integrals wrt an arbitrary measure on an arbitrary set of that cardinality (the Haar measure on the compact group $(\mathbb Z/2\mathbb Z)^{2^\omega}$ is the first that springs to my mind). $\endgroup$ – Emil Jeřábek Apr 24 '16 at 9:17
  • $\begingroup$ @EmilJeřábek, I think that one finds the phrase "formal expression" rather than "formal definition" for your second meaning, but I think we are in agreement about the post. $\endgroup$ – Joel David Hamkins Apr 24 '16 at 12:19
  • $\begingroup$ @Emil and Joel. Emil, your second definition of "formal" is exactly what I have in mind and I agree with Joel that I should better talk about "formal expression" instead of "formal definition". Could also talk about "symbolic expression". I will fix this. $\endgroup$ – Fabrice Pautot Apr 25 '16 at 8:23
  • $\begingroup$ @Emil and Todd. Emil, you are referring to my question Sums, series, integrals. What's next? where I give the "concrete" example of the hypothetical sums/means over "Beth2 terms" (I know that's a serious abuse of terminology) I have in mind: "functional mean images" that are symbolically defined in mathoverflow.net/questions/232043/…. I can't see the link with the Haar measures you mention, but if they are supposed to be related, I'd like to know. $\endgroup$ – Fabrice Pautot Apr 25 '16 at 8:25
  • $\begingroup$ @Emil and Todd, I'm not even 100% sure that it is logically required to well-define those "functional mean images" I have in mind in order to well-define and compute the probability distribution(s) of a deterministic signal, which are what I'm really interested in. But without any input about those important probability distributions, I just told to myself that I should better ask first about the hardcore mathematical objects on which they seem to rely. $\endgroup$ – Fabrice Pautot Apr 25 '16 at 8:34
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It is not clear that "formally defined" as you use it helps anyone here. What do you want to do with the construct? What notion are you trying to capture? If measure spaces, functional analysis, and calculus of variations do not hold some of the truth you seek, what are you seeking?

My take on your question is that you want something that is different from measure on a function space, yet you still want to use the word "sum" . Is that really appropriate? If you are considering probability calculations, aren't you already in possession of a probability measure over which to operate? What meaning (setting aside temporarily issues of existence) could you assign to a mean value of a collection of functions evaluated at x_0? Shouldn't it be 0, as you will have -f(x_0) in your collection for each occurrence of f(x_0)?

There should be cases existing in measure theory or functional analysis where one looks at a measure on a function space. Do a search for those terms, study the results, and then come ask your question and tell us why your study did not answer them. At this point I just see a presentation I don't understand of an entity with probabilistic connotations that has some associated symbols that you say constitute a formal definition, but does not give me something to work with. And you ask me if it exists? I respond with "So what if it does or doesn't? I can't use it or relate it to things I understand, so at present I can't help resolve that question."

A more acceptable version of the question would be: " I have the following context where I am supposed to integrate over a function space rather than over a finite product of the reals. Where can I find examples of how that is used? " . You might get it closed with a comment of "look at Banach spaces and their operators", but at least that would be progress.

Gerhard "Not Uncountably Many Steps Backward" Paseman, 2016.04.23.

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    $\begingroup$ To supplement this answer: Regarding measures on function spaces, I'd recommend taking a look at Da Prato's Introduction to Infinite-Dimensional Analysis and Bogachev's Gaussian Measures. (Be warned: This topic is not for the faint of heart.) $\endgroup$ – Christian Clason Apr 24 '16 at 8:57
  • $\begingroup$ @Christian. Thank you very much for the references. I'm already aware that R^R and other sets of cardinal at least Beth2 I'm considering here are "nasty" and "problematic" from the measure-theoretic point of view. $\endgroup$ – Fabrice Pautot Apr 25 '16 at 8:03
  • $\begingroup$ @Gerhard for the full line of reasoning/context underlying my question, please see: mathoverflow.net/questions/236527/…. For a short version: mathoverflow.net/questions/232043/… $\endgroup$ – Fabrice Pautot Apr 25 '16 at 9:05
  • $\begingroup$ @Gerhard, as pointed it out by Emil and Joel, I should better talk about "symbolically defined" or "formal expressions" instead of "formally defined". Corrected $\endgroup$ – Fabrice Pautot Apr 25 '16 at 9:07
  • $\begingroup$ @Gerhard: thank you Beth2 times! You are the first to be kind enough to consider seriously this hypothetical "mean functional image". Should it be equal to 0 for any $x_0$, it would be perfectly well defined. I would feel good but this would have serious consequences in dynamical system theory. But why should 0 plays any particular role? In particular, any reason why the hypothetical "functional mean image" of $x_0$ cannot be equal to $x_0$ for instance? $\endgroup$ – Fabrice Pautot Apr 25 '16 at 9:17
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    $\begingroup$ I think ultimately you want help in defining/understanding/discussing a set of concepts. MathOverflow is not setup for discussion, it is for asking and answering specific questions. Lay out a cogent framework, ask a detailed question about a specific point, and the community may respond and may even provide requested references. The format of your question suggests to me you want a discussion as much as an answer. That's great, but MathOverflow is not the place for such. Hopefully the hints I and Christian gave help you. Gerhard "Ask Me About Not Discussing" Paseman, 2016.04.25. $\endgroup$ – Gerhard Paseman Apr 25 '16 at 16:13
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    $\begingroup$ Wait a minute --- If this previously undefined object were to be defined to be identically zero, it would have serious consequences in dynamical system theory? Cool! I hereby define it to be identically zero. I always wanted to be a famous dynamical system theorist. $\endgroup$ – Steven Landsburg Apr 25 '16 at 17:01
  • $\begingroup$ @Gerhard: the specific question is here: mathoverflow.net/questions/232043/…. Unfortunately, I don't get any answer. That's why, in a second step, I've asked for the existence of the low-level mathematical objects ("functional mean images") on which my high-level objects (probability distributions of deterministic signal) seem to rely. $\endgroup$ – Fabrice Pautot Apr 26 '16 at 6:29
  • $\begingroup$ @Steven, if "my" "functional mean images" would be identically zero, then the marginal joint prior probability distribution of a deterministic signal would be identically zero. In particular, it would be invariant under permutation of the chronological order. It can't be so. Please see my specific question: mathoverflow.net/questions/232043/… $\endgroup$ – Fabrice Pautot Apr 26 '16 at 6:34
  • $\begingroup$ @Steven. After 416 views of my specific, fundamental dynamical systems-theoretic question mathoverflow.net/questions/232043/…, I haven't received any answer nor comment. Absolutely nothing. That's why I really start to fear that my questions and the mathematical objects I've introduced (some strange looking functional integrals) might pertain to a branch of mathematics (= probabilistic dynamical systems theory???) that is at least not so well known, if it ever exists. $\endgroup$ – Fabrice Pautot Apr 26 '16 at 9:51
  • $\begingroup$ @Steven. Assuming that MO members is a representative population of the whole mathematics, by Laplace's rule of succession the probability of the probabilistic theory of dynamical systems (to be already known) is, as of today, equal to $1 - (420+1)/(420+2) = 0.0023$. $\endgroup$ – Fabrice Pautot Apr 26 '16 at 10:11
  • $\begingroup$ @StevenLandsburg, if you want to be a famous dynamical system theorist, study the Collatz conjecture. Gerhard "That's A Famous Dynamical System" Paseman, 2016.04.26. $\endgroup$ – Gerhard Paseman Apr 27 '16 at 4:30
  • $\begingroup$ @GerhardPaseman. Yes, together with Feynman path integrals, the Collatz conjecture provides another good example to illustrate the present situation. In both cases, we are facing easy to formulate, even elementary problems that nobody can solve (or is willing to solve). Even, mathematics is not yet ready for such problems (Erdös). The main difference, however, is that CJ doesn't look to be a fundamental question per se. On the contrary, marginalizing dynamical systems is absolutely fundamental because it is the first step towards a probabilistic theory of deterministic signal. $\endgroup$ – Fabrice Pautot Apr 27 '16 at 7:25
  • $\begingroup$ @GerhardPaseman. The fact that nobody (total number of views on MO+MSE+PSE is 666 as of today) can provide any sharp, accurate answer to such an elementary and fundamental question makes me believe that the probability of the probabilistic theory of dynamical system I am considering is in fact well below the one given by Laplace. $\endgroup$ – Fabrice Pautot Apr 27 '16 at 7:34
  • $\begingroup$ @GerhardPaseman. At the beginning, I was expecting somebody to answer me something like "the probability distribution of a deterministic signal is obviously equal to this" or "marginalizing state-space equations is addressed in this paper", "marginalizing phase spaces is addressed in that paper" or "your functional integrals are in fact well known as this in that theory". Now, I really fear I will never get any answer like this but I still hope I'm wrong. $\endgroup$ – Fabrice Pautot Apr 27 '16 at 7:46

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