Celebrating Learnable Mathematics
The Work of Professor F. William Lawvere
In introducing Professor F. William Lawvere in ‘Celebrating Bill Lawvere and Fifty Years of Functorial Semantics’, Professor Kimmo I. Rosenthal noted how Professor F. William Lawvere attends to questions from undergraduate students with as much attention and care as he does to questions from senior colleagues. I am very fortunate to experience Professor F. William Lawvere’s generosity of spirit in response to my very first question and our correspondence continued for decades since then, which gave me the courage that I too can contribute, however miniscule it maybe, to the advancement of mathematics.
To provide a perspective, in stark contrast to the Newtonian ‘body’ idealized as existing in-and-of-itself, according to modern category theory, every object is an object of a category of objects, which aligns with our categorical perception. What I find particularly fascinating in Professor F. William Lawvere’s papers, textbooks, and correspondence is pointers to how much there is to understand following discussions of mathematical advances in a manner readily accessible to total beginners, much of it is available in the recently launched The Lawvere Archives, thanks to Madam Fatima Lawvere’s unwavering commitment to make science commonsense.
Functorial semantics of algebraic theories is revolutionary in that it made the theory of a category of mathematical objects a mathematical object. Note that theory, prior to functorial semantics, was a list of statements, or models, or presentations of models, just to list a few. Reminiscent of Einstein highlighting the kinship between scientific and ordinary thinking, Professor F. William Lawvere brought into figural salience the propinquity between mathematical knowing and knowing in general along with that of abstraction of concepts (but cognitive science shies away from it all; doesn’t even include Mathematics in Cognitive Science logo, which is beyond the reach of my puny brain). Along these lines, Professor F. William Lawvere showed that geometry provides its own foundation, while duly crediting all those who contributed and led to every contribution of his that transformed mathematics beyond recognition. One lesson that I learned from Professor F. William Lawvere, which I value more than anything else, is that science is not a spectator-sport; it requires conscious participation in scientific practices, which hopefully crystalizes into a theory of the practice that can then be used to guide the practice.
With logic as the algebra of parts (cf. calculating the number of subsets of a set A using the formula 2|A|, where 2 = {false, true} is the truth value set of the category of sets), we find that the truth value object or subobject classifier of a mathematical category can be calculated based on the theory of the category, and the rest of the corresponding logic follows from operations on the thus calculated truth value object. Recognizing the limitations of subobject classifier, Professor F. William Lawvere points out the need to develop a broader objective logic, by considering arbitrary morphisms instead of the representability of monomorphisms (parts) considered in calculating the more familiar narrow objective logic.
Along with the objective logic that is intrinsic to the universe of discourse, we have objective number theory (for example, 1 + 1 = 2 corresponds to the arithmetic of the category of sets, while 1 + 1 = 1 corresponds to the arithmetic of the category of pointed sets, such as time-line as a set of points, with origin as its distinguished point).
While duly acknowledging the value of subjective logic of inferring statements from statements, Professor F. William Lawvere pointed out the significance of the laws of rational passage between concepts constituting propositions. In this context, it must be noted that naturality understood as ‘Becoming consistent with Being’, not only holds it all together, but also makes science (in the sense of ever-proper alignment of reason with experience) possible, thereby making the so-called ‘unreasonable effectiveness in mathematics in physical sciences’ mute.
More than anything else, mathematics is not so much so about quantities, but about qualities, which (both extensive and intensive qualities) Professor F. William Lawvere abstracted from the ongoing mathematical practice as a special case of cohesion. The mathematical motivation for the study of qualities is clearly visible in the sense every quantity is a quantity of something, the qualities of which need to be mathematically characterized. Then there are integration problems, the significance of which was highlighted by none other than Newton as ‘compounding after resolution’ or synthesis-after-analysis, which makes it clear that contemporary reductionist analysis is at best the first half of the practice of science. Along these lines, one long overdue deliverable is the refinement of our understanding of the relation between space and time, beyond that given in velocity, which can accommodate the qualitative attributes of space (e.g., stick together) and time (cf. urge to move). Last but not least, in view of the limitations of Cantor’s almost structureless sets in serving as a background to model cohesively varying everyday extended objects, we need a background that’s totally devoid of qualities: formless-and-essenceless mathematical construct for contamination-free representations.
Summing up, The Lawvere Archives is not only a repository of published papers, but also a window into the making of science, the significance of which for the training of scientists can’t be underestimated.