# Mathematical Notation and its importance

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?

• 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 4 '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 5 '14 at 9:15
• Many thanks for the comment and the book suggestion! – Bruno Alessi Feb 6 '14 at 12:03
• You are welcome ! O posso dire : prego ? – Mauro ALLEGRANZA Feb 6 '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

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.

• 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 6 '14 at 12:04

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?

• 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 5 '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. – Slade Feb 5 '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 5 '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...) – Slade Feb 5 '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. – Slade Feb 5 '14 at 16:22

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.

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.