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The possible properties and applications of some mathematical objects have been described far before their rigorous mathematical definition. Some of them even had a seemingly paradoxical description which prevented them from existing at the time.

An iconic example is the imaginary number, $i$, which was originally described as $\sqrt{-1}$. However, further advancements in the field and exploring new perspectives towards the subject eventually led to a rigorous non-contradictory mathematical definition of $i$. A definition that serves as a formal introduction of the (already existing) concept to the mathematical literature while satisfying all of its required properties.

Similar attempts have been conducted along the lines of providing a solid formal basis for seemingly paradoxical concepts such as negative probability or sets with a negative number of elements (see also multisets):

Loeb, D., Sets with a negative number of elements, Adv. Math. 91, No. 1, 64-74 (1992). ZBL0767.05005.

Another contemporary example of such paradoxical mathematical objects which are pending for a rigorous definition is the so-called "field with one element". It naturally appears in various occasions and relates different concepts together. For instance, a group can be seen as a Hopf algebra over the "field with one element". See also this related MO question for further information.

Question. I am looking for more examples of such deep and useful mathematical concepts with contradictory descriptions which are not rigorously defined yet (or at least have no widely accepted formal definition). References to their possible applications are also welcome.

Remark. I would like to emphasize the initial paradoxical description of the possible answers to this question. The important point is that the natural appearance of paradoxical objects often indicates the urgent need for a deep expansion of our mathematical worldview or an essential improvement of the foundations.

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marked as duplicate by Noah Snyder, Yemon Choi, Mark Sapir, Andrés E. Caicedo, Stefan Waldmann Aug 14 '18 at 9:07

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ The so-called field with one element ? $\endgroup$ – Sylvain JULIEN Aug 12 '18 at 12:01
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    $\begingroup$ The key fact is that it's not just a singleton but a yet not well defined object behaving much like a field of characteristic one, i.e. such that $ 1+1=1 $, with quite rich combinatorial interpretation. It was first considered by Tits and has been further explored by Connes and Consani, among others. $\endgroup$ – Sylvain JULIEN Aug 12 '18 at 12:11
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    $\begingroup$ @MonroeEskew It's more complicated than that. $\endgroup$ – Noah Schweber Aug 12 '18 at 13:01
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    $\begingroup$ Quantum field theory? $\endgroup$ – Michael Bächtold Aug 12 '18 at 13:54
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    $\begingroup$ Are ordinals an historical example? (Burali-Forti paradox) $\endgroup$ – Monroe Eskew Aug 12 '18 at 16:14
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Path integrals are a standard example. Rigorous path integrals can be formulated in many cases but there is still no completely general approach to path integrals that satisfies both physicists and mathematicians.

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This paper of Parisi makes a case for the following two objects: $r$-dimensional Euclidean space for $r$ not an integer, and the space of linear transformations on that space (= $r\times r$ matrices with real entries). Parisi argues that these are ideas which ought to have meaning, but do not yet.

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The self-avoiding walk was famously analyzed by de Gennes using a generalization of the Ising model known as the $N$-vector model. By taking the limit as $N\to 0$, he obtained results for the self-avoiding walk. However, there is currently no known way of making this argument rigorous, in part because $N$ is an integer and it is unclear how to rigorously allow $N\to 0$. The meaning of the $N$-vector model when $N$ is not an integer may be thought of as a "paradoxical" object. For more information, see Section 2.3 of The Self-Avoiding Walk by Neal Madras and Gordon Slade.

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    $\begingroup$ I can't help noticing that all three answers so far refer to objects introduced by physicists... $\endgroup$ – Carlo Beenakker Aug 12 '18 at 21:10
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    $\begingroup$ @CarloBeenakker : This could be a matter of culture. For example, we are still pending a fully satisfactory construction of mixed motives, but we mathematicians tend to phrase our ignorance in terms of precisely stated conjectures, and we shy away from boldly presenting a non-rigorous "paradoxical" argument. Physicists seem more uninhibited. $\endgroup$ – Timothy Chow Aug 12 '18 at 22:01
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    $\begingroup$ @CarloBeenakker what about dimensional regularization, that is, integration over $\mathbb R^d$ with $d\in\mathbb C$? :-P $\endgroup$ – AccidentalFourierTransform Aug 12 '18 at 22:59
  • $\begingroup$ This is almost a special case of the answer by @noah schweber $\endgroup$ – lcv Aug 12 '18 at 23:30
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Grothendieck surmised the theory of motives 50(?) years ago, but people are still trying to figure out what it is. I'm not sure if there are "paradoxes" involved though.

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Fractional derivatives has been defined and found useful in some contexts.

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    $\begingroup$ What is contradictory with them? $\endgroup$ – Sebastian Bechtel Aug 13 '18 at 13:44
  • $\begingroup$ @SebastianBechtel Well, the standard definition of the derivative, which is well-known to the general mathematical audience, physicists, engineers is essentially integer-indexed. First derivative = speed, second = acceleration, etc. It is mind-blowing to think that something can be 'in between'. $\endgroup$ – mathreader Aug 14 '18 at 3:43
  • $\begingroup$ @mathreader mind blowing, but legitimately useful. While sometimes we get strange results in between (differentiating power functions gives a pole at origin due to Gamma function for example) within carefully chosen regions I've found fractional derivatives useful in researching differential operators. In fact, I remember reading a paper that mentioned a version satisfying a version of Gauss-Lucas.... Perhaps I could find it. $\endgroup$ – Brevan Ellefsen Aug 14 '18 at 4:12
  • $\begingroup$ @SebastianBechtel as far as what is contradictory, a big issue is that there is no "fractional derivative" that works the way we want. Some are simple, some are complicated, some differentiate power series the way we want, some satisfy Liouvilles Theorem, etc. The problem is that none of them gives us everything we want, so we have to find a way to choose. For a comparable problem, consider the extention of integer Tetration to complex powers. We have multiple candidate proposals that all seem to be roughly similar, but determining which to choose has been... Difficult. $\endgroup$ – Brevan Ellefsen Aug 14 '18 at 4:14
  • $\begingroup$ @SebastianBechtel in fact, it wasn't till this last year (iirc) that we proved there exists a unique complex Tetration preserving the fixed points of complex logarithm (details can be found, e.g., on Wikipedia page). Returning to fractional differentiation, entire tomes have been written on the subject, with more proposed definitions than I can count on my fingers - each seems nice in its own way, but nothing as of yet stands out to make a clear "winner" (not to say some definitions aren't more useful than others... Far from it) $\endgroup$ – Brevan Ellefsen Aug 14 '18 at 4:18
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The Dirac delta function is a "function" which is zero almost everywhere but integrates to one. No function can possibly behave like that, but it nevertheless seemed like a useful notion. It was eventually made rigorous by the theory of distributions (or generalized functions).

It shows up paradoxically if you try to figure out what the identity element for the convolution operator. You can quickly determine that no such element exist, but if it were to it would have such and such properties. Said properties are exactly the defining properties of the Dirac Delta.

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    $\begingroup$ The question asks for objects 'which are not rigorously defined yet' (my italics). In other words, it seems to be asking for present examples, not historical ones. As you point out, we do now know how to rigorously define the Dirac delta function. $\endgroup$ – HJRW Aug 14 '18 at 7:02
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From what I have read, the concept of fractional order calculus still does not have a rigorous basis. It has a somehow paradoxical nature similar to that of complex numbers mentioned by the OP, and the topic is as old as the calculus itself.

Although the fractional integration can be well-defined using the Cauchy's formula, but fractional differentiation does not have a rigorous definition because of the difficulty of dealing with the initial conditions.

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  • $\begingroup$ I should say that when I started to write this answer, @mathreader's answer wasn't there and I got too much distracted to the point that writing these lines took me 1 hours! So sorry for a seemingly duplicate answer $\endgroup$ – polfosol Aug 13 '18 at 12:07
  • $\begingroup$ There are several different definitions of this idea. en.wikipedia.org/wiki/… $\endgroup$ – Monroe Eskew Aug 13 '18 at 12:48
  • $\begingroup$ @MonroeEskew that's why I said it's not well-defined $\endgroup$ – polfosol Aug 13 '18 at 12:50
  • $\begingroup$ Sure it is. Just well-defined a few different ways. $\endgroup$ – Monroe Eskew Aug 13 '18 at 12:52
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    $\begingroup$ I'm skeptical that there is an underlying concept. $\endgroup$ – Monroe Eskew Aug 13 '18 at 12:58

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