Yesterday we discussed the limit of the process of compounding interest. As we move through even shorter periods, from semi-annual to quarterly to monthly compounding and so forth, and as we approach an infinity of infinitesimal periods, i,e. continuous compounding, we find the number 2.71828 arises to stare us in the face as a limit. This was Jacob Bernoulli's result. So why is the number called Euler's number, and sometimes just e? Did Euler name it for himself? And, if so, isn't that an injustice? Shouldn't it be called b? Well, Bernoulli found 2.71828... but he neglected to name it. Gottfried Leibniz, the man recognized as THE founder of calculus in the German speaking world, seems to have called it "b" in letters he wrote in 1690 and 1691. Perhaps this was a recognition of Bernoulli's role. More important, though, Bernoulli neglected to explore the properties of this number outside of banking. It was Euler (pictured above) who defined it as the...
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Again, I'm looking at Fiona Cowie's book on innatism, past and present. My last post on this book discussed Descartes in connection with what Cowie calls the "special faculties" hypothesis in innatism, the view that certain features of our mind are modular, so that learning X is an qualitatively different thing from learning Y. Cowie distinguishes this from what she calls the "mystery" hypothesis in innatism, aka rationalism. Rationalists, like empiricists for that matter, "understand that both experience and our innate endowment play critical roles" in our development, learning, etc. The first issue that divides them, as we have seen, is whether the innate endowment is to be understood in a unitary or in a modular manner. The second issue that divides them is the idea of mystery, the idea prominent in the writings of the classical rationalists that there is no natural explanation of these special faculties, that they resist empirical invest...
I'm working my way carefully through Fiona Cowie's book, WHAT'S WITHIN? NATIVISM RECONSIDERED (1999), a book I've only skimmed until now. I believe I've mentioned it here a time or two on the basis of the skimming and its reviews. The book, which began life as a doctoral dissertation, is a look at Fodor and Chomsky in particular, and their revival of elements of classical rationalism, especially with regard to a priori ideas. Today I'll focus on an early point, within Cowie's historical discussion. She asks: what exactly was the classical debate, the one featuring Descartes and Leibniz on one side, figures such as Hobbes and Locke on the other, really about? It is not easy to pin in down. The empiricists sometimes wrote as if the rationalists were saying that a baby comes into the world already knowing, say, what a triangle is, or what constitutes a prime number. Babies clearly don't. And the old rationalists denied this is what they meant. D...
Plotinus wrote that, yes, the sensible world is a mere imperfect copy of the intelligible world, but he also wrote against the pessimism of the gnostics, against the idea that the sensible world is a bad thing. He wrote thus: “what more beautiful image could there be? After the fire of the intelligible world, what better image of it could there be than our fire? What earth, outside of the intelligible earth, could be better than ours?" In short: yes, we live in a cave looking at shadows, but the cave is not at all a bad place to be. Those who aren't philosophers (most of the species, on Plotinian premises) can never get outside the cave, but that doesn't make their fate too dire. They get to look at all those fascinating shadows on the wall. Meanwhile for those few who can sometimes get out of the cave and look at the real world ands even glimpse the sun, those times are of necessity brief. We'll have to return to the cave. We can resign ourselves to th...
If a believer in God is going to have a theodicy, that is, a measured effort to “justify the ways of God to man,” he is going to have to go in one of three directions. There are only three. The problem is this. If God is all-powerful, then He can bring an end to evil. If God is ideally benevolent, then He wants to bring an end to evil. So: why is there evil? Three answers: you can choose to remain silent and regard the question as an unanswerable mystery (which Job learns to do at the end of the OT book bearing his name). Or you can define “all-powerful” in a way that solves the problem. Or you can define “benevolent” in a way that solves the problem. The problem is created by two constraints: that of power and that of goodness. Although no theodical authors would put it this way, some of them define “power” down and others define “goodness” down, loosening the one constraint or the other. In the late 17th century, Leibniz famously defined "power" down. ...
The story of another candidate for numberhood, the infinitesimal, is even stranger than the story of the irrationals or the imaginaries. The infinitesimal is the limit of a process, where the method stipulates that the end of that process can never be reached. Consider: can a single mathematical point have a slope? Our intuitive answer, trained by Euclid, is: no. A point is pure position. It isn’t a line, it isn’t even a tiny part of a line, so it can’t have a slope! Can there be a smallest possible line? How short can a line get and still have a slope! That’s a question that Euclid taught us not to ask. It is akin to suggesting that we can have two adjacent points. If we could have two adjacent points, then they would presumably constitute the smallest possible line and they would have a slope. But we can’t. Between any two points, properly speaking, infinity of other points can fit. In the seventeenth century it was common for the mathematicians at its cuttin...