The Limits of Mathematical Consciousness

       There are three kinds of people, the saying goes -- those that can count, and those that can't.  And that right there might well describe the limits of mathematical consciousness.  End of post.

       But let's hear from some other people before we let the subject go.  Albert Einstein, no slouch in the discipline of mathematics, has been quoted as saying, "Not everything that counts can be counted, and not everything that can be counted counts."

       Which is a good saying, I think.  Don't you?

       But there is a catch here: maybe Einstein didn't say it.  Researchers into the matter of correctly attributing quotations suggest that the quotation should be credited to a professor of sociology, William Bruce Cameron, who said something very like this in publications back in the 1950s and 1960s.   Should we accept the word of a sociologist on matters mathematical?

       Or maybe it was a physician who said this first.  Or a religious writer.  You can see the results of current research in a post at the website, Quote Investigator.  Maybe we should establish a quote of our own, that would go something like this: "Not everything that has been quoted is properly attributed, and not every idea that is worthy of attribution comes from an authoritative quote."

       You can quote me.  Or say it yourself.  Or find that someone has beaten us to the punch-line.

       Back to mathematics.  Benjamin Disraeli, a British prime minister of the 19th century, is famously quoted as saying, "There are three kinds of lies:  lies, damned lies, and statistics." (Statisticians, pollsters, accountants, economists, politicians, census takers, and actuaries, take note.)

       Once again, it is possible that Disraeli didn't say this, even though Mark Twain thought he did.  And neither of them was a professional mathematician, and thus may have lacked certain refinements of mathematical consciousness; but each of them had important things to say -- and think -- and the history of human consciousness would be poorer if either of them had not lived.

       John von Neumann, a twentieth century mathematician and physicist and logician of remarkable breadth and brilliance, said,  "There's no sense in being precise when you don't even know what you're talking about."

       Mathematical consciousness has enjoyed especial prestige since the days of Sir Isaac Newton and his disciples, for whom God was, among other things, a consummate Mathematician.  From that day to this Mathematics has been more or less the Handmaid of Science, or the Queen of the Sciences, depending upon whom you listen to.

       But there are limits.  It is by no means certain that all things, or all important things, are quantifiable; or that they "can be mathematically modeled"; or even that they can be mathematically symbolized.  Like life.  Or love.  Or liberty.

       Or dimensionality.  Or thought.  Or consciousness.  Or personality.  Or beingness.  Or even the wagging of a dog's tail.

       I think that mathematical consciousness, once it is self-aware, begins to wonder about meta-mathematical consciousness, and . . . much else.  It is good to welcome other voices, other possibilities.

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Your thoughts are most welcome, even if they are limited!  (Or unlimited.)