Abstract
Recent years have seen a growing acknowledgement within the mathematical community that mathematics is cognitively/socially constructed. Yet to anyone doing mathematics, it seems totally objective. The sensation in pursuing mathematical research is of discovering prior (eternal) truths about an external (abstract) world. Although the community can and does decide which topics to pursue and which axioms to adopt, neither an individual mathematician nor the entire community can choose whether a particular mathematical statement is true or false, based on the given axioms. Moreover, all the evidence suggests that all practitioners work with the same ontology. (My number 7 is exactly the same as yours.) How can we reconcile the notion that people construct mathematics, with this apparent choice-free, predetermined objectivity? I believe the answer is to be found by examining what mathematical thinking is (as a mental activity) and the way the human brain acquired the capacity for mathematical thinking.
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Notes
One aspect of my argument is clearly amenable to scientific investigation right now, namely the extent to which the sensation of “exploration of an external (though abstract) reality” that accompanies doing mathematician is prevalent among the professional mathematical community. My claim that it is widespread, indeed dominantly so, is based on many casual conversations with other professional mathematicians, together with reading a number of published self-reports. While I find this input sufficiently compelling to motivate this speculative paper, it would be useful to have a properly conducted statistical study.
A positive outcome to such a study would of course establish such an “external reality” as a social fact. That is all that this paper assumes.
I am using the word “circuitry” in a somewhat metaphorical fashion, to refer to established neural activity patterns.
This is essentially the argument presented by Bickerton (1995).
Dreaming is an interesting intermediate case, since dreams do appear to start spontaneously, in many non-humans as well as in humans. Despite considerable research, however, little is known about dreaming. It appears to involve random combination of parts of activation chains that have already been produced during awake activities. The repeat of parts of “meaningful” activation chains, which were once initiated by physical stimuli, may be what gives dreams their (recollected) content, even though there is a strong element of random firing involved. But since our only conscious knowledge of our dreams is the recollection we have when we wake up, any meaningfulness may be illusory. And we have no way of knowing whether non-human dreaming, say by a cat or a dog, involves any mental activity meaningful to that animal. Given the problems involved in trying to understand what goes on in dreaming, it is far more likely that a greater understanding of thinking will cast light on the nature of dreams than that discoveries about dreams will help us understand thinking.
A second feature of mathematics that makes it hard is the degree of rigor required in its reasoning processes. Precise reasoning is not something for which our brains evolved. But we need to be careful in drawing conclusions from this observation. Precise, formal reasoning is not required for mathematical discovery. Rather, its purposes are verification of things already discovered (or perhaps suspected) and convincing others of the truth of those discoveries.
The need for formal verification is a direct consequence of the nature of mathematical discovery. Trial and error, guesswork, intuition, and conversations with others can go on for days, months, even years, with key steps often being carried out while the mathematician is either asleep or thinking about something else. Although this process need not be mere haphazard stumbling—for a good mathematician it can be highly focused and efficient—it can nevertheless generate errors. Formal proofs are the final (and in principle totally reliable) safeguard against false “discoveries.”
References
Bickerton, D. (1995). Language and human behavior. Seattle, WA: University of Washington Press.
Devlin, K. (2000). The Math Gene: How mathematical thinking evolved and why numbers are like gossip. Basic Books.
Hadamard, J. (1949). The psychology of invention in the mathematical field. Princeton University Press.
Lakoff, G., & Johnson, M. (1980). Metaphors we live by. University of Chicago Press.
Lakoff, G., & Nunez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books.
Rumelhart, D., & McClelland, J. (1986). Parallel distributed processing: Explorations in the microstructure of cognition. MIT Press.
Weyl, H. (1944). David Hilbert and his mathematical work. Bulletin of the American Mathematical Society, 50, 612–654.
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Devlin, K. A Mathematician Reflects on the Useful and Reliable Illusion of Reality in Mathematics. Erkenn 68, 359–379 (2008). https://doi.org/10.1007/s10670-008-9105-2
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DOI: https://doi.org/10.1007/s10670-008-9105-2