History of visualization in science and math - where may I read of it?

Question

I am looking to inquire into the role of visualization in sciences and math and wish to read about the history of visualization in those fields and also wonder where I might read about the epistemology of visualization in math and physics.

Answer 1

I can give some suggestions in regard to visualization in mathematics. As far as I know, not much has been written about it.

There is a book by James Robert Brown, Philosophy of Mathematics-A Contemporary Introduction to the World of Proofs and Pictures$^1$, which devotes many pages to the subject of the use of images and the theory of visualization in mathematics.

This is a book of philosophy of mathematics from a Platonist point of view. But it deals with a contemporary version of Platonism, that goes beyond the original Platonism and also beyond the Platonism of some prominent mathematicians of the XX century, as Kurt Gödel and Godfrey H. Hardy.

And it develops an original idea of Platonism that, according to the author, is capable of dealing with many questions of philosophy of mathematics with a wider and more flexible approach, also with respect to other philosophy of mathematics as formalism and structuralism.

In particular, with regard to the issue of mathematical visualization, Brown writes:

Platonism, more than any other account of mathematics, is open to the possibility of an endless variety of investigative techniques. Proving theorems in a traditional way is certainly one method of establishing new mathematical truths, but it needn't be the only way. [...]

Platonism can be [...] liberating for mathematical research.

[...] what other means might there be? [...]and, especially, what about diagrams and pictures? Besides traditional proofs, these not standard techniques may also bear much fruit. $^2$

As we can see, this is an original view of the use of images in mathematics, which now enter the very core of mathematics, proofs, whereas they are usually considered a tool and a support of mathematical theories that have been built elsewhere and with different tools.

Brown's book devotes, in particular, two chapters to mathematical images: Chapter 3 - Picture Proofs and Platonism, and Chapter 9 - Proofs, Picture and Procedures in Wittgenstein, but the issue of mathematical images goes through the whole book.

His treatment is mainly philosophical, but there are plenty of mathematical examples and also historical notes about the subject. And, of course, there is also a bibliography.

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From a historical point of view, we can say that there has been a 'tension' between mathematics and images, as diagrams and pictures. Whereas the use of pictures and diagrams is widespread in mathematics texts, diagrams have been often considered possibly misleading.

In particular, some diffidence toward diagrams, graphs and pictures has accompanied 'waves' of rigorization of mathematics.

Joseph Louis Lagrange, who was very attentive to mathematical rigour, remarks in his Mécanique Analityque (1788) :

No figures will be found in this work. The methods I set forth require neither constructions nor geometrical or mechanical arguments, but only algebraic operations, subject to a regular and uniform procedure (1788, Pref.) $^3$.

In XIX century, in particular during the so-called second rigorization, of which Bernard Bolzano was a pioneer and carried on by Karl Weierstrass and his school, there was a new attempt to expunge pictures and geometrical consideration from mathematics.

Pictures were considered not rigorous, for example in the issues concerning the definition of the continuity of functions. For instance, in the case of the Intermediate Value Theorem, of which Bolzano gave a purely analytic proof, avoiding the geometric self-evident idea of a continuous function as a function whose graph is a one-piece line, a line that can be drawn without lifting the pencil from the paper.

The present-day, of Weierstrassian origin, definition of a continuous function also expunges this visual idea: in particular, according to the present-day definition, there are continuous functions whose graph cannot be drawn without lifting the pen from the paper. $^4$

The removal of visual and geometrical aspects continued during the arithmetization of analysis, with the exigence of establishing the foundations of mathematical analysis, and in particular of the theory of real numbers also on numbers, without geometrical aspects. And, in general, it was peculiar of the logical and foundational research on set theory and foundations of mathematics, the search for mathematical roots, of the first part of the XX century.

But there wasn’t only a stance against visualization. At the same time, at the end of the XVIII century and during the XIX century, an important field of mathematical analysis, complex analysis, was intertwined with geometry and images.

Wessel, Argand and Gauss, all recognized that complex numbers could be given a simple, concrete, geometrical interpretation as points in the plane: the mystical quantity $a+ib$ should be viewed simply as the point in the x-y plane having cartesian coordinates $(a,b)$ […] the reputation of Gauss ensured wide dissemination and acceptance of complex numbers as points in the plane[…] now existed some way of making sense of these numbers - that they were now legitimate objects of investigation […]. Riemann’s ideas, in particular, would simply not have been possible without prior knowledge of the geometry of the complex plane.$^5$

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The history of rigorization continued during the XX century, in particular with Bourbaki, a highly influential collection of French mathematicians who stressed axiomatics and rigour in the development of mathematics.

Jean Dieudonné, one the most important mathematicians of the Bourbaki group, was an arch-enemy of pictures and graphs.

In the preface to his Foundations of Modern Analysis(1969) Dieudonné states:

a strict adherence to axiomatic methods, with no appeal whatsoever to “geometric intuition”, at least in the formal proofs: a necessity which we have emphasized by deliberately abstaining from introducing any diagram in the book.$^6$

An exclamation by Jean Dieudonné entered the annals of history of mathematics. During a conference about the didactics of mathematics in 1959, the Colloque de Royaumont, a workshop of ten days whose objective was to introduce a reform of the teaching of mathematics in schools, Dieudonné, with regard to the teaching of geometry and pictures, cried:

À bas Euclide! À mort le triangle!$^7$

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$^1$ ^{James Robert Brown, Philosophy of Mathematics-A Contemporary introduction to the world of Proofs and Pictures, (2008) second ed., Routledge.}

$^2$ ^{ibid. p. 15}

$^3$ ^{In Brown, cit. p. 199.}

$^4$ ^{According to the contemporary definition of continuous function, a function defined only at isolated points is continuous, but its graph is evidently composed of isolated points.}

$^5$ ^{Tristan Needham, Visual Complex Analysis, 1997, Oxford University Press, pp. 2-3.}

$^6$ ^{In Brown, cit. p. 199.}

$^7$ ^{(“Down with Euclid! To death the triangle!”). See Maurice Marshal, Bourbaki, 2017, Edition Belin/Humensis, p. 222. Also https://matematica.unibocconi.it/articoli/anche-l%E2%80%99insegnamento-della-matematica-viene-coinvolto-una-revisione-di-obiettivi-contenuti-}