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You can see how mathematical notation evolved during the last centuries here.

I think everyone here knows that a bad notation can change an otherwise elementar problem into a difficult problem. Just try to do basic arithmetics with roman numbers, for example.

As a computer programmer I know that in some situations programming language notation plays a critical rule because some algorithms are better expressed in a particular language than in other languages even considering they all have the same basis: Lambda Calculus, Turing machines, etc

The linguists has their so-called Sapir–Whorf hypothesis which "...holds that the structure of a language affects the ways in which its respective speakers conceptualize their world, i.e. their world view, or otherwise influences their cognitive processes."

Then, I ask: is there any field in Math that studies Math's notation and its influence for good or for bad in Math itself?

Modifying the fragment on the paragraph above: is it possible that the notation, the symbols and the language used in Math affects the ways in which Mathematicians conceptualize their world and influences their cognitive processes?

Mark S.
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Bruno Alessi
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  • I think that symbolism has a power of abstraction; our way of thinking about "abstract" things is (at least) influenced by the way we symbolize them: how I can "speak of" an object that I not perceive if I cannot symbolize it ? So, a "good" symbol may help has, while a "bad" one may inhibit us. The anthropologist Lévi-Strauss said that the myths are «bonne à penser»; so are the symbols. – Mauro ALLEGRANZA Feb 04 '14 at 16:12
  • You can see Albrecht Heeffer & Maarten Van Dyck (editors), Philosophical Aspects of Symbolic Reasoning in Early Modern Mathematics (2010). – Mauro ALLEGRANZA Feb 05 '14 at 09:15
  • Many thanks for the comment and the book suggestion! – Bruno Alessi Feb 06 '14 at 12:03
  • You are welcome ! O posso dire : prego ? – Mauro ALLEGRANZA Feb 06 '14 at 12:03
  • I wonder How could they even work using full sentences to describe equation. That is extremely hard and inefficient! – Trismegistos Feb 12 '14 at 14:20
  • possibly related: http://math.stackexchange.com/questions/555895 and http://math.stackexchange.com/questions/93922/ and http://mathoverflow.net/questions/42929/ – Mark S. Feb 15 '14 at 04:11
  • There is no notation in the first link (the page has changed). – Pedro Sep 07 '16 at 12:42

3 Answers3

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On a quite different tack, you might well be interested in Mohan Ganesalingham's The Language of Mathematics: A Linguistic and Philosophical Investigation (Springer 2013).

The author is an outstanding mathematician (Senior Wrangler, no less), and has a degree in linguistics, and now works in computer science. The book is based on a prize-winning thesis. I mention those facts in case the word "philosophical" in the sub-title puts you off! Mohan seriously knows his stuff.

Peter Smith
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  • Math, Linguistics and Philosophy is an interesting combination. Many thanks for the suggestion - I know many books but not this one, nor that one Mauro mentioned. – Bruno Alessi Feb 06 '14 at 12:04
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In my opinion, this is one of the exciting promises of intuitionistic mathematics and topos theory.

By discarding the law of the excluded middle $\neg\neg P\implies P$ (of course, we can add it back later if we want), much more of the structure of our axioms becomes evident in our theorems, because we can no longer label arbitrary statements as true-or-false.

For example, in topos theory, one no longer speaks of "the" real numbers, but of "a" real numbers object in a topos. When the topos is $Set$, nothing special happens. But in a different setting, we may have a countable real numbers object, or nontrivial subobjects without points. My understanding is that there are also topoi for which every function $\mathbb{R}\to\mathbb{R}$ is differentiable (which should be a relief to any physicists who pretend this all the time).

I like this view because it makes certain properties of "the" real numbers—its cardinality, for example—appear to be artifacts of the language of sets, which relies upon a firm notion of membership. But since the vast majority of real numbers cannot be pinned down in any meaningful sense, it is not hard to argue that treating $\mathbb{R}$ as a set is, at the very least, a choice of perspective.

And this can actually be useful—the now-classic quantum physics paper What is a Thing? has argued that the standard $\mathbb{R}$ is inadequate for non-classical theories of physics. After all, if the number of particles in a system is dependent on how we measure the system, then why should we expect anything in the universe to behave like a set?

Andrew Dudzik
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  • It is not quite true that "we can add it later if we want", in reference to the axiom of choice. Some the results of intuitionist mathematics rely on existence of elements that are neither nonnegative nor nonpositive; a case in point is the counterexample to the extreme value theorem found in Troelstra and van Dalen's famous book. – Mikhail Katz Feb 05 '14 at 15:38
  • @user72694 This is like saying that we cannot pass from rings to fields because it would contradict the existence of DVRs. In topos theory, the double negation subtopos corresponds to taking only the Boolean sheaves, and hence says nothing about sheaves that are not Boolean. – Andrew Dudzik Feb 05 '14 at 15:51
  • That's in fact a very good analogy. Once you have incorporated rings with divisors of zero into your framework, you can't complete your objects to become fields. Similarly, if the background logic is intuitionistic, typically you will have to discard some of your theorems if you want to "complete" the logic to classical. – Mikhail Katz Feb 05 '14 at 16:07
  • @user72694 Yes, but who said that passing from intuitionistic to classical logic has anything to do with completion? If $\mathcal{T}$ is a topos, then the Boolean part $\mathcal{T}{\neg\neg}$ is a subobject of $\mathcal{T}$. This is analogous to passing to a subspace, not a completion. But this has nothing to do with classical or intuitionistic logic; the same happens when we add any axiom to any system. (and there _is a perfectly reasonable functor from rings to fields...) – Andrew Dudzik Feb 05 '14 at 16:13
  • @user72694 It's worth pointing out that an intuitionistic existence theorem over a topos $\mathcal{T}$ is always connected to a topos $\mathcal{S}\to\mathcal{T}$. So it is no surprise that we cannot "compose" theorems with the subtopos $\mathcal{T}{\neg\neg}\to\mathcal{T}$. What we can do, however, is examine the base change $\mathcal{S}{\neg\neg}\to\mathcal{S}$. Objects that still exist classically are exactly those that lie in the essential image. – Andrew Dudzik Feb 05 '14 at 16:22
  • This is very interesting but one does not need topos theory to have different models of the real line. Troelstra and van Dalen develop everything in essentially a set-theoretic framework with background logic being intuitionistic. I think more editors here could follow your comments if you stick to more familiar frameworks. – Mikhail Katz Feb 05 '14 at 16:33
  • @user72694 Familiar to whom? A relative topos is an intuitionistic model; they are not different. One wishing to avoid category theory can let $\mathcal{T}=\mathcal{Set}$ and $\mathcal{S}$ be a locale, which covers all the basic examples I'm familiar with. I know of no framework in which it is not already understood that the addition of axioms does not preserve $\exists$. – Andrew Dudzik Feb 05 '14 at 16:45
  • What I was pointing out is that the comment that LEM "can always be added back later" is controversial. Errett Bishop always sought to present his mathematics in such a way as to appear compatible with classical mathematics, while others (notably Feferman) have pointed out that Bishop is finessing the issues, while books such as Troelstra and van Dalen explicitly deal with situations where one cannot turn the clock back. This was discussed in this article. – Mikhail Katz Feb 05 '14 at 16:48
  • @user72694 It's not controversial to anybody who has studied topos theory. Here is an analogous statement in algebraic geometry: it is worthwhile to consider $\mathbb{P}^n$ instead of $\mathbb{A}^n$, as we can restrict later if necessary. There are numerous complaints that one could make, e.g. "But $\mathbb{P}^n$ doesn't have enough global sections!" This is clearly just nitpicking; nobody thinks that you can prove a theorem about global sections on $\mathbb{P}^n$ and then use it on $\mathbb{A}^n$. Yet the various functors are well-understood. Math has no "clock" to turn back. – Andrew Dudzik Feb 05 '14 at 16:56
  • I agree with the thrust of your answer but was merely pointing out that some of your comments can be misinterpreted by a reader not sufficiently well versed in topos theory (so your "of course" seems a bit tongue-in-cheek :-)). Incidentally, the concept of "discarding LEM" is not really well-defined. There are several distinct intuitionistic logics that become equivalent if one introduces LEM. – Mikhail Katz Feb 05 '14 at 16:58
  • @user72694 That is true about discarding LEM. I don't see what misinterpretation is possible, though. In order to prove that some intuitionistic model exists, I believe that it is necessary to use classical logic anyway; otherwise, how can you define consistency? Within a given proof, there is no issue with adding axioms later instead of right away. This, I thought, was the whole point of the intuitionists—heck, I've heard ultrafinitists express their comfort with a suitably-formulated classical logic. – Andrew Dudzik Feb 05 '14 at 17:18
  • Errett Bishop would have a heart attack if he heard this. The idea of using classical logic to attain constructive aims will not have pleased him at all. Certainly LEM is true over suitable finite domains and even some infinite ones provided they are suitably decidable. – Mikhail Katz Feb 05 '14 at 17:21
  • ...More specifically, I don't think Bishop would be interested in the issues of consistency. As Goedel showed, consistency can only be shown in the context of yet a larger system, which leads to infinite regress. To Bishop all such issues would be in the realm of what he politely called "idealistic mathematics" and behind the scenes referred to as "fundamentalist mathematics". – Mikhail Katz Feb 05 '14 at 17:32
  • @user72694 How does one even assign meaning to a statement like "there exists a model of theory $\mathcal{T}$" without passing to a larger logical system than the one in which $\mathcal{T}$ is formulated? I think that you are unnecessarily conflating peripheral philosophical issues with the core, uncontroversial theory. Note that even the article you linked refers to "pre-LEM" and "post-LEM" reasoning. – Andrew Dudzik Feb 05 '14 at 18:24
  • I think Bishop was happy with the "existence" of the natural numbers without having to invoke models. You probably wouldn't find the term model in Bishop even if you use a torchlight. – Mikhail Katz Feb 05 '14 at 18:30
  • @user72694 Your original point was that the extreme value theorem has a counterexample in some intuitionistic model of the reals (and yes, Troelstra and van Dalen do speak of models). Why raise the issue in the first place if you are just going to reject it? If we don't care about models, then there is no argument to be had. – Andrew Dudzik Feb 05 '14 at 18:36
  • Well, I don't reject models. Bishop does. He is one famous constructivist. So when you want to make statements about intuitionism and/or constructivism you need to take him into account. – Mikhail Katz Feb 05 '14 at 18:41
  • @user72694 Do you have some kind of a point? I was taking into account model-less views when I made my original statement; models only came up because your objection requires them. There are no counterexamples in pure intuitionism, only examples and non-examples. – Andrew Dudzik Feb 05 '14 at 20:12
  • Nice! I'm starting a Physics course this year and will keep this in mind, specially the last paragraph. I know Math is essential to Physics as well as very interesting by itself. I hope my Math knowledge grows and I can start answering questions too, not only asking. Now, I'm thinking in what way I can mark this question as answered, since you and the other two guys answered it nicely. Must I choose only one or more than one? – Bruno Alessi Feb 06 '14 at 12:10
  • @User-33433, let me provide an additional example of why it is misleading to claim that "LEM can always be added back later". Synthetic differential geometry a.k.a. Smooth infinitesimal analysis (of Lawvere, many other people, and most recently John Bell) is formulated in a category-theoretic framework with intuitionistic logic. Because the logic is intuitionistic, the theory can exploit nilpotent infinitesimals that cannot be proved to be nonzero, but nonetheless can be exploited to do calculus with as in the good old days... – Mikhail Katz Feb 06 '14 at 13:43
  • ...Re-introducing LEM into the theory forces these infinitesimals to vanish and the whole theory collapses. In other words, intuitionistic logic is an essential feature of the theory. That was my kind of a point. – Mikhail Katz Feb 06 '14 at 13:44
  • What, precisely, does this answer have to do with mathematical notation? – user642796 Feb 06 '14 at 19:28
  • @user72694 Let me know when you have an objection based on mathematics, not survey articles. It is tiring being told on the one hand that I shouldn't use Lawvere-style categorical reasoning because you don't know topos theory, only to be told much later that my way of thinking is at odds with him and the same theory. – Andrew Dudzik Feb 06 '14 at 20:56
  • @BrunoAlessi You must choose one. I promise not to be offended (and the other answer arguably is more directly relevant). – Andrew Dudzik Feb 06 '14 at 21:02
  • @ArthurFischer In the broad scheme of things, category theory is not an enormous departure from modern mathematics, linguistically speaking. But it is still somewhat of a departure. I'm not going after the philosophical issues of language here, just a single, practical move from one foundation to another. A proof in categorical terms, simply put, is notated differently from the same proof in set-theoretic terms. I think of topos theory as being about the concrete mathematical consequences of this syntactical choice. – Andrew Dudzik Feb 06 '14 at 21:09
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Here is an interesting recent point of view:

Foundations of Science March 2014, Volume 19, Issue 1, pp 1-10 Script and Symbolic Writing in Mathematics and Natural Philosophy

Maarten Van Dyck, Albrecht Heeffer

Abstract

We introduce the question whether there are specific kinds of writing modalities and practices that facilitated the development of modern science and mathematics. We point out the importance and uniqueness of symbolic writing, which allowed early modern thinkers to formulate a new kind of questions about mathematical structure, rather than to merely exploit this structure for solving particular problems. In a very similar vein, the novel focus on abstract structural relations allowed for creative conceptual extensions in natural philosophy during the scientific revolution. These preliminary reflections are meant to set the stage for the following contributions in this volume.

See http://link.springer.com/article/10.1007/s10699-012-9310-y

Another recent article that has a bit too much psychology in it for my taste but may be of interest to you is the following:

January 2013, Volume 190, Issue 1, pp 3-19

Mathematical symbols as epistemic actions

Helen De Cruz, Johan De Smedt

Abstract

Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols are not only used to express mathematical concepts—they are constitutive of the mathematical concepts themselves. Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare brains. Using case-studies from the history of mathematics and from educational psychology, we argue for an intimate relationship between mathematical symbols and mathematical cognition.

See http://link.springer.com/article/10.1007%2Fs11229-010-9837-9

Mikhail Katz
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