Open letter to Lakoff about his book on math

Dear Prof. Lakoff

A short note to say how much I am enjoying "Where Mathematics Comes From". It is providing closure for me.

In the 1980's, I fell in love with math during a college course based on the very foundations-conscious Calculus text by Spivak and although I enjoyed the rigor and beauty, I could not help but feel a bit uneasy at the hoops one had to jump through for the proofs to work, all this in the name of eliminating intuition.

I eventually ended up on the "applied" side, to the regret of my wonderful teachers, because I felt much more at ease with the way things seemed to work out intuitively. Successive approximations to non-linear functions using the university mainframe seem to do the trick of finding solutions to things I was curious about. Infinitesimal approaches, simplex algorithms and Fourier series implemented in WATFOR to solve hard differential equations numerically were simple but magical approaches to understanding. I was gratefully standing on Turing and Church's shoulders.

Anyhow, over the years I kept looking for insights as to why the pure math left me a bit uneasy since I still admired the beauty of it, (like pining for a high-school sweetheart) in all kinds of places including David Foster Wallace's overly baroque book on the math of infinities (yikes).

Your book reconciled the comfort of the applied and the beauty of the formal for me. There is a cognitive continuum after all, between the purists and us engineering/computing folk.

It connected the dots, clarifying the things that seemed like bridges by grounding them. By focusing on the metaphors for infinity it helped in categorizing things that were both computing and theory, but that could only exist because of computing: compiler theory, the Aho-Ullman stuff derived from Chomsky. I can now see why there was all this cognitive dissonance when I was learning Analysis.

For example, the space-as-points versus space-containing-points argument you and Nunez make is beautiful and could be applied to the philosophy of physics as well. The notion of particle comes to mind. I think Feynman the intuitionist in physics is a counterpart to Poincare in the world of math. His explanations of energy quanta and fields address the analogous notions of space and points in the world of physics, where it seems some still cling to the Platonic reality of math. The intuitionists, like Poincare, were very courageous and insightful as well as demonstrably able to make new discoveries as they sidestepped the Gottingen school's rejection of intuitive geometry and Cantor's objectification of infinities. Set theory is nice but infinity is just NaN.

I had read Kline's history of math some years ago and although I agreed with his view that a rift in math was created at the time when intuition was relegated to a sideline, there seemed to be something missing, much more to say. Your work with Nunez closed that loop for me.

I am now curious and encouraged to find out more about any research to make rationals (the real Reals :-)) the basis for discretization, as well as links between foundations work and Chomsky's formal grammars. I agree as the book infers that terminology in math is key to disambiguation - but technical terms like Real and continuous are too deeply entrenched of course. Maybe intro calculus courses should begin with a review of those specialized semantics. Like lawyers learning contract terms...

I am also curious about reactions to your work from the math community (rather than by the pedagogical folks) since I think the arguments are revolutionary and deeply original. I looked cursorily on the Web but found little of substance in reaction or support of the ideas in the book. Are mathematicians looking into closing the gaps you identify? Did you get much substantive opposition to your views? Finally, kudos for the other wonderful work on political discourse. Sorely needed. I wish I could be young enough again to enroll as a student of your faculty, but I will continue to follow your research with great interest nonetheless.

Cognitive Philosophy

There seems to be a movement afoot - long overdue - to embody the foundations of intelligence and philosophy. It is partially old news, the heroic AI of the 50's, 60's and 70's has been discounted despite our benefiting from the results of that research in today's mainstream technology: neural nets, machine speech recognition, handwriting recognition, natural language translation, cars that drive themselves, and all kind of other low-level intelligent (unconscious?) activities simulated by programs.

I think that it is the Minsky flavour of AI (now re branded cognitive science), the one grounded in logic, seems to have fallen out of favour -since the fruits of that work did not rise up to the expectation of the "I" in AI; an AI based only on logic, a formal game with symbols that pretended to be words. How can that be smart without the observer having to make all kinds of allowances and assumptions?

This stuff was subject to sometimes acrimonious debate for a while, until technology and big data made many of the arguments moot.

The good news is that there is now a strong movement towards the recognition of the role of the perceptual environment and its embodiment in animals like us as part of the equation.

This realization may be due to the Web, indirectly, since it should be now be obvious that the seeming "intelligence" of the Google search would not exist without the "environment" of all the Web sites, which are of course interactions with people's brains (in fact there was a very simple AI program that used "Google distance" as a way to measure metaphorical strength between concepts. For example "Castro" and "cigar" scored pretty high on that scale).

So now there is a movement, a possibly serendipitous program of research coming together to ground AI in the real-world of experience, and some of the people that I think are leading this effort align themselves along a somewhat elegant but odd symmetry of ideas.

Let me explain.

There is David Gelernter, the computer scientist and columnist who wrote "The Muse in the Machine" and "Mirror worlds". These books made me want to go back and look at AI again. In fact I ran back. His arguments are so cogent, so seemingly simple and common-sensical that I almost missed their radical novelty. He states that data and the links between data is the key to intelligence. These links are often emotional gradients, in the sense that logic is not always what binds ideas; it is similar emotional affect!

The brain associates memories based on the closeness of the emotions they evoke, the sensory input parallels that occurred as the memory is created. The whole model, not just the idea...how you felt, how cold or hot or tired or dizzy you were. This links data to environment so elegantly that it is a mystery to me why no-one has picked up on this earlier. He breaks the syntax/semantics dichotomy very deeply. Also, Gelernter has very conservative political views - right of centre. I am noting this now so as to make my strange symmetry argument a bit further down. Bear with me.

There is George Lakoff, the cognitive linguist. His work with Johnson on embodied philosophy and with Nunez on the origins of math, takes the empirical evidence around the links between our brain wiring and the perception of the world and uses it as a foundation for philosophy.

Basic brain mechanisms are considered to be foundational and ideas grow as hierarchies of metaphor (i.e. links).

Metaphor becomes a key mechanism for cognition. Similarly to Gelernter,  Lakoff et al propound that metaphors are based on evolutionary and environmental constraints, not some platonic logic ideal. No such thing as the separation of  mind from the world of things. These guys are existentialist scientists. Finally! The rampant positivism and Platonism of the English speaking world, the Whitehead/Russell school, is giving ground. Merleau-Ponty and Husserl are now more visible in the world of scientific cognition.

Lakoff is an ex-student of Chomsky. He seems to disagree with the flavour of Chomsky' thesis of hard-wired syntax in the brain. He believes syntax and semantics to be more of a spectrum of complexity in cognitive biology than a differentiator between humans and other animals. Also he is a non-neo-con, a strong and vocal opponent of  right wing politics as it is played in the US today, and although I hesitate to slot him with Chomsky on the political front, they are on the same side of a hypothetical centre line.

So we have right and a left wing thinkers on cognition who fundamentally disagree in the political realm, the realm of people relations, nations, influence, and power. That is a healthy and good thing. This lends support to the notion that there is a grounding in "objective" reality for this thinking on cognition, it is much more than opinion.  (A slim argument, I know, but hey, this is my blog, I can say what I want.)

Aside from the political thing, I want to continue to talk about is the cognition thing. There is another thinker/scientist that seems to be breaking new ground along those same lines.

He is a neuro-scientist, one who gets his hands dirty opening up craniums, working with brain surgeons. William H. Calvin. I read two books of his: "Conversations with Neil's Brain" and "How Brains Think, Evolving Intelligence Then and Now". His ideas centre around the notion that cognition is an initially subconscious Darwinian process among neural bundles that remember perceptions, integrate them and ultimately model the environment perceived by the body, and that the successful "species" of memories/emotions (let's call them cognitions) are the ones that we call conscious, the ones that rise to the top of the noise of cognition, to differentiate themselves as our inner voices.

His research fundamentally agrees with the models put forth by Gelernter and Lakoff. Like Lakoff, he believes that most cognition in the brain is fomented through unconscious processes. Like Gelernter he finds that memories are linked through non-logical processes. He shares with Lakoff a deep knowledge of Chomsky's ideas on innate language ability and like Lakoff disputes the simplicity of these models, without denying the human brain's ability to parse language. He also talks of metaphor, not surprisingly, since all this is supported by empirical evidence at the chemical and behavioural level.

So I am very happy that this stuff is happening. I tended to get hot under the collar when I read "classic" texts by Minsky or Pinker on intelligence. They seemed simplistic and smug. I was always perplexed by the lack of attention given to Grey Walter's ideas which preceded but anticipated Gelernter's. I wanted 'The Muse in the Machine" to be discussed more, but to my chagrin, Penrose's strange books on quantum brain hypotheses got more press.

Let's get back to empirical research, let's continue the program of work that will ultimately bring philosophy and psychology back into science.The foundational thinkers list above should also include Turing. Even Penrose mentions his "other work". He did mathematical research on molecular biology (plants mostly) and how stem cells can become complex structures. His mathematically-based hypotheses on these matters were validated experimentally just this year in the UK. Ultimately this theoretical work may prove to explain some of the brain mechanisms underlying the work Calvin and others are doing.

There are conceptual layers to all this analysis, from the bottom-up: the molecular, the cellular, the systems (bundles of neurons evolving), and the cognitive and behavioral, leading to metaphor, logic and science, including physics, on to philosophy, morality  and politics. The gaps needs to be filled so that we have a continuum of understanding. A New Science in the Vico sense. Something we made and understood that we made, but that is grounded in experience.

(I just found an earlier post that was wishing this stuff would happen, and so here we are.)