In 1734, Voltaire wrote
Qu’il y a des carrés d’infini, des cubes d’infini, et des infinis d’infinis, dont le pénultième n’est rien par rapport au dernier? Tout cela, qui paraît d’abord l’excès de la déraison, est en effet, l’effort de la finesse et de l’étendue de l’esprit humain, et la méthode de trouver des vérités qui étaient jusqu’alors inconnues.
What a beautiful thought expressed by a mind in reverence of Newton and his invention of differential calculus!
That there should be squares of infinity, cubes of infinity, and infinities of infinities, of which the penultimate is nothing in relation to the last? All of that, which at first seems like an excess of reason gone mad, is in fact, the effort of the perspicuity and extent of the human mind, and the method of finding truths that were until then unknown.
What is equally stunning is just how many forms of representations humanity has invented.
I used the foregoing as the opening quotation in my honors thesis in Psychology (1990), “A computational investigation of the evolution of vision“, because I wanted to stress that concepts, taxonomies, symbolisms, languages, syntaxes, grammars, formalisms, representations and representational schemes, are individually and certainly all together, more general and fundamental than facts.
Psychology is a very empirical science, and it was (and still is) rare for an honour’s or Ph.D. thesis to be theoretical. Rarer still are psychology theses that explore the space of possible designs (as my thesis did). So, I had to make a case not merely for my results, but for my very aims.
(Incidentally, in 1990, the Department of Psychology of McGill University kindly offered me admittance to their Ph.D. program. I asked Prof. Shultz, one of the two profs who wanted me in their lab, if anyone had ever defended a theoretical dissertation in his department. He said no! I didn’t want to have to defend the very process of doing cognitive science properly. So, I turned down McGill and accepted a Commonwealth Scholarship and FCAR to study at what was then Europe’s best school of cognitive science, Sussex’s, under Aaron Sloman. Also then present were Andy Clark, Maggie Boden, Phil Agree, and other luminaries. With all due respect to McGill, I have never regretted my decision.)
In order to understand the human mind, one must not merely ask Nature questions in the form of experiments (or other types of empirical study). One also has to explore, on an a priori basis, the space of possible designs, designs which She may or may not have ever produced. And that requires, more fundamentally, that one have formalisms for expressing designs. My honour’s thesis explored such formalisms.
Consider that thinking and more generally “mentation” are not most fundamentally constrained by the models we use to interpret the psychological and physical world—though, to be sure, these constructions limit as well as enable us. They are constrained formally by the concepts and representations that we use, which are, themselves, theoretical.
Voltaire realized that without differential calculus, there is an upper limit to how we can think about physical phenomena. Calculus, while redefining the very concept of limit, does not remove our representational limitations; but it certainly extends them.
Cognitive science, with Artificial Intelligence as its core discipline, is an advance on vanilla psychology because it develops and embraces powerful representational schemes, such as “connectionism“, which, incidentally also leverages calculus.
This is not to say that the lay reader need to wield such formalisms herself. But she may like to know that theoreticians whose work she delves into use mind-stretching concepts.