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stoutfellowAt this point I have to recap a little bit of philosophical history. I do so with some hesitation; philosophy is far from being one of my strong suits, and what I say below should probably be taken with more than one grain of salt. However, I believe it is correct in its main outlines, and it is actually relevant to certain nineteenth-century developments in mathematics.
One of the perennial problems of philosophy is that of knowledge: how is it possible to know things? To be certain?
There are two traditional lines. One attempts to construct a priori truths, and build from there; the other bases itself on experience: the evidence of our senses, the things we perceive. In the modern era, Descartes is an exemplar of the former. His famous Cogito, ergo sum is the first step: "The fact that I think is grounds for believing that I exist." (This is actually disputable, and has been disputed by thinkers ranging from Gautama to Marvin Minsky. But I digress.) John Locke rejected this; on his view, the mind, prior to receiving sense-data, is a blank slate, a tabula rasa. All reasoning, he declared, begins there.
David Hume, a generation or so after Locke, presented the philosophers of his day with a seemingly intractable dilemma, by examining the notion of causation. The cause-and-effect relation, he declared, is not at all derivable from a priori principles, since it is possible to conceive of worlds in which the same cause gives rise to different effects from those we are familiar with. On the other hand, it is not deducible from sense experience either, since causation is not itself perceivable; we can perceive that this event is associated with that, but we cannot perceive that the one causes the other. How, then, can we be certain about these matters? (I am assured that Hume was not arguing against the idea of causation itself, but only against the idea that we can be certain about it.)
It was Immanuel Kant who took up Hume's challenge. He distinguished the noumenal world, the world as it really is, from the phenomenal world, the world we perceive. Granting that we cannot really know anything about the former, he suggested that we can know things about the latter. The very way that our minds work, the way we perceive things and the way we process those perceptions, force certain conclusions on us; and these we may as well take as true. (Metaphorically, one might ask, "How can I be certain that the next thing I see will be blue?" Response: "I am wearing blue spectacles!") Kant then outlined various truths of this sort.
Since one of the longstanding boasts of mathematics is its claim to certain truth, it is not surprising that Kant included mathematical statements in his list. Since the premier example of mathematical reasoning had for centuries been Euclid's Elements, it is not surprising that Kant pointed to Euclidean geometry as part of that guaranteed truth. Since Kantian philosophy and its descendants dominated thinking in Germany, and indeed in most of Europe, for many years, it is not surprising that this had consequences for mathematics - but that is a story for another day.
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One of the perennial problems of philosophy is that of knowledge: how is it possible to know things? To be certain?
There are two traditional lines. One attempts to construct a priori truths, and build from there; the other bases itself on experience: the evidence of our senses, the things we perceive. In the modern era, Descartes is an exemplar of the former. His famous Cogito, ergo sum is the first step: "The fact that I think is grounds for believing that I exist." (This is actually disputable, and has been disputed by thinkers ranging from Gautama to Marvin Minsky. But I digress.) John Locke rejected this; on his view, the mind, prior to receiving sense-data, is a blank slate, a tabula rasa. All reasoning, he declared, begins there.
David Hume, a generation or so after Locke, presented the philosophers of his day with a seemingly intractable dilemma, by examining the notion of causation. The cause-and-effect relation, he declared, is not at all derivable from a priori principles, since it is possible to conceive of worlds in which the same cause gives rise to different effects from those we are familiar with. On the other hand, it is not deducible from sense experience either, since causation is not itself perceivable; we can perceive that this event is associated with that, but we cannot perceive that the one causes the other. How, then, can we be certain about these matters? (I am assured that Hume was not arguing against the idea of causation itself, but only against the idea that we can be certain about it.)
It was Immanuel Kant who took up Hume's challenge. He distinguished the noumenal world, the world as it really is, from the phenomenal world, the world we perceive. Granting that we cannot really know anything about the former, he suggested that we can know things about the latter. The very way that our minds work, the way we perceive things and the way we process those perceptions, force certain conclusions on us; and these we may as well take as true. (Metaphorically, one might ask, "How can I be certain that the next thing I see will be blue?" Response: "I am wearing blue spectacles!") Kant then outlined various truths of this sort.
Since one of the longstanding boasts of mathematics is its claim to certain truth, it is not surprising that Kant included mathematical statements in his list. Since the premier example of mathematical reasoning had for centuries been Euclid's Elements, it is not surprising that Kant pointed to Euclidean geometry as part of that guaranteed truth. Since Kantian philosophy and its descendants dominated thinking in Germany, and indeed in most of Europe, for many years, it is not surprising that this had consequences for mathematics - but that is a story for another day.
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