Following the previous posts on bhāvanā, bringing-into-being, and its role in meditation, literature, and human creative activity in general, I couldn’t help but relate some of those ideas to mathematics.
Many (pure) mathematicians are content to take the mathematical structures they explore as givens, which they can figure out or manipulate in interesting and sometimes profoundly beautiful ways. This is a naïve version of Platonic realism, which grants to mathematical structures an objective existential status that lies beyond human beings. (This also means that sentient alien species should have exactly the same mathematical ideas as we do, that the Vulcans would accept that Euclid’s axioms generate the same results as us, and so on.)
Philosophers of mathematics who are formalists of various stripes hold instead that mathematics comes down to playing games with symbols: pushing arrows and boxes and Greek and Hebrew characters around based on well-defined rules. (Cue Wittgenstein.) They reject the idea that mathematicians “discover” mathematical structures; rather, they formulate new rules and new symbols and manipulate them.
While it may certainly seem from the outside that this is all mathematicians do, and while Western models of logic separate formal syntax and formal semantics in a way that seems to encourage this line of thought, it does not gel with my personal experience of actually doing proofs. Seldom can a real proof be hit upon simply by pushing symbols around on a piece of paper. At least for me, thinking about a hard mathematical problem involved trying very hard to “see” what was going on behind the symbols: symbols barely came into it. Strangely enough, the harder the proof, the more I thought I was seeing something that was already there! And even in those cases when one does merely shuffle things around, there is a crucial psychological difference between staring at a bunch of symbols on a page and hitting the Eureka moment. The proof is complete, I would argue, only with the latter. (This is not a “proof” that formalism is wrong, but merely an observation that it doesn’t fit with the phenomenology of at least some mathematicians.)
So where does bhāvanā come into the picture here? I would like to suggest (without proof, hehe) that what makes a proof a proof is precisely the fact that when(ever) it is understood correctly, it reliably and unfailingly brings into being in our minds a mathematical truth, in a manner that is at least intersubjectively valid, if not objectively. A proof is the means by which a particular mathematical end (a fact, a theorem, a lemma, or whatever) is attained.
I have been vaguely inclined towards this manner of thinking ever since I read one of the greatest math books written in recent times in my opinion, Tristan Needham’s Visual Complex Analysis. Needham takes perhaps the most aesthetically remarkable branch of modern mathematics and offers a fabulous tour of its key features and structures in a manner that emphasizes visual and geometric thinking over the algebraic. (That is, he encourages you to prove things not by pushing symbols on paper but by visualizing, rotating, and dilating mathematical structures.) Given my prior bias towards visualizing mathematical structures, this book has been particularly enjoyable to read. (Perhaps my favorite exercise in visualization is the one in which I had to “see” the complex logarithm multifunction twisting and lifting the complex plane into an infinite helix.)
This process of visualizing a mathematical object is both deeply personal and yet objectively available. Two people who visualize a mathematical object will both agree on its key characteristics and its relevant properties, and may yet visualize it in ways that differ quite dramatically (and yet inexpressibly) from each other. To me, this situation bears a thought-provoking resemblance to Hindu/Buddhist meditative exercises in which devotees are asked to bring-into-being a particular deity in their minds, and are usually given elaborate visual descriptions of the deity’s characteristics to aid them in the process. Two different devotees may thus both bring-into-being very different versions of the same deity in their own minds, while yet agreeing fully on all of the key features possessed by this deity. The former half allows them to “take ownership” of the deity, in a sense; the latter half lets them participate in a shared conversation with others about the deity. Of course, by comparing meditative exercises with mathematical proofs, I intend to make neither religious claims about mathematical entities nor mathematical claims about religious entities!
Many (pure) mathematicians are content to take the mathematical structures they explore as givens, which they can figure out or manipulate in interesting and sometimes profoundly beautiful ways. This is a naïve version of Platonic realism, which grants to mathematical structures an objective existential status that lies beyond human beings. (This also means that sentient alien species should have exactly the same mathematical ideas as we do, that the Vulcans would accept that Euclid’s axioms generate the same results as us, and so on.)
Philosophers of mathematics who are formalists of various stripes hold instead that mathematics comes down to playing games with symbols: pushing arrows and boxes and Greek and Hebrew characters around based on well-defined rules. (Cue Wittgenstein.) They reject the idea that mathematicians “discover” mathematical structures; rather, they formulate new rules and new symbols and manipulate them.
While it may certainly seem from the outside that this is all mathematicians do, and while Western models of logic separate formal syntax and formal semantics in a way that seems to encourage this line of thought, it does not gel with my personal experience of actually doing proofs. Seldom can a real proof be hit upon simply by pushing symbols around on a piece of paper. At least for me, thinking about a hard mathematical problem involved trying very hard to “see” what was going on behind the symbols: symbols barely came into it. Strangely enough, the harder the proof, the more I thought I was seeing something that was already there! And even in those cases when one does merely shuffle things around, there is a crucial psychological difference between staring at a bunch of symbols on a page and hitting the Eureka moment. The proof is complete, I would argue, only with the latter. (This is not a “proof” that formalism is wrong, but merely an observation that it doesn’t fit with the phenomenology of at least some mathematicians.)
So where does bhāvanā come into the picture here? I would like to suggest (without proof, hehe) that what makes a proof a proof is precisely the fact that when(ever) it is understood correctly, it reliably and unfailingly brings into being in our minds a mathematical truth, in a manner that is at least intersubjectively valid, if not objectively. A proof is the means by which a particular mathematical end (a fact, a theorem, a lemma, or whatever) is attained.
I have been vaguely inclined towards this manner of thinking ever since I read one of the greatest math books written in recent times in my opinion, Tristan Needham’s Visual Complex Analysis. Needham takes perhaps the most aesthetically remarkable branch of modern mathematics and offers a fabulous tour of its key features and structures in a manner that emphasizes visual and geometric thinking over the algebraic. (That is, he encourages you to prove things not by pushing symbols on paper but by visualizing, rotating, and dilating mathematical structures.) Given my prior bias towards visualizing mathematical structures, this book has been particularly enjoyable to read. (Perhaps my favorite exercise in visualization is the one in which I had to “see” the complex logarithm multifunction twisting and lifting the complex plane into an infinite helix.)
This process of visualizing a mathematical object is both deeply personal and yet objectively available. Two people who visualize a mathematical object will both agree on its key characteristics and its relevant properties, and may yet visualize it in ways that differ quite dramatically (and yet inexpressibly) from each other. To me, this situation bears a thought-provoking resemblance to Hindu/Buddhist meditative exercises in which devotees are asked to bring-into-being a particular deity in their minds, and are usually given elaborate visual descriptions of the deity’s characteristics to aid them in the process. Two different devotees may thus both bring-into-being very different versions of the same deity in their own minds, while yet agreeing fully on all of the key features possessed by this deity. The former half allows them to “take ownership” of the deity, in a sense; the latter half lets them participate in a shared conversation with others about the deity. Of course, by comparing meditative exercises with mathematical proofs, I intend to make neither religious claims about mathematical entities nor mathematical claims about religious entities!
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