“It’s not objectively out there in the world.” And two versions of set theories can come up with opposite results. Why? Watch the video.
From this piece located at the publications page of the International Computer Science Institute. “Mathematical models help describe reality, but only by ignoring its inherent integrity.” Computers work on binary logic and the world is full of ‘noise.’ Hence computers, and mathematical models for that matter, can only approximate reality by eliminating that noise.
“Can a bunch of bits represent reality exactly, in a way that can be controlled and predicted indefinitely? The answer is no, because nature is inherently chaotic, while a bunch of bits representing a program can never be so, by definition.”
Which leads us to ask: “Are our mathematical models just a desperate, failed attempt to de-noise an otherwise very confusing, extremely blurred reality?”
So yes, math and computers are quite useful as long as we keep the above in mind instead of assuming they reveal reality as it is. And as long as we also search for that noisy humanity in the spaces between binary logic, which will never be revealed by math or computers alone.
The nature of math came up in our embodied cognition discussion. Here is a presentation by Dehaene on the topic followed by comments from The Reality Club: George Lakoff, Marc Hauser, Jaron Lanier, Rafael Núñez, Margaret Wertheim, Howard Gardner, Joseph Traub, Steven Pinker, Charles Simonyi. A few brief, edited Lakoff excerpts follow from that discussion. Note that this is a discussion from 1997, so a lot of confirming science has happened since then.
” [Dehaene] has made it clear that our capacity for number has evolved and that the very notion of number is shaped by specific neural systems in our brains. […] We understand the world through our cognitive models and those models are not mirrors of the world, but arise from the detailed peculiarities of our brains.”
“Mathematics is not ‘abstract’, but rather metaphorical, based on projections from sensory-motor areas that make use of ‘inferences’ performed in those areas. The metaphors are not arbitrary, but based on common experiences: putting things into piles, taking steps, turning around, coming close to objects so they appear larger, and so on.”
“Dehaene is right that this requires a nonplatonic philosophy of mathematics that is also not socially constructivist. Indeed, what is required is a special case of experientialist philosophy (or ’embodied realism’). […] Such a philosophy of mathematics is not relativist or socially constructivist, since it is embodied, that is, based on the shared characteristics of human brains and bodies as well as the shared aspects of our physical and interpersonal environments. […] On the other hand, such a philosophy of mathematics is not platonic or objectivist. Consider two simple examples. First, can sets contain themselves or not? This cannot be answered by looking at the mathematical universe. You can have it either way, choosing either the container metaphor or the graph metaphor, depending on your interests.”
“Dehaene is by no means alone is his implicit rejection of the Computer Program Theory. Distinguished figures in neuroscience have rejected it (e.g., Antonio Damasio, Gerald Edelman, Patricia Churchland). Even among computer scientists, connectionism presents a contrasting view. In our lab at the International Computer Science Institute at Berkeley, Jerome Feldman, I, and our co-workers working on a neural theory of language, have discovered results in the course of our work suggesting that the program-mind is not even a remotely good approximation to a real mind.”
Pinker of course also comments and takes issue with Lakoff’s depiction of him. Dehaene also responds to the comments at the end.