P. Bernays has pointed out on several occasions that, since the consistency of a system cannot be proved using means of proof weaker than those of the system itself, it is necessary to go beyond the framework of what is, in Hilbert's sense, finitary mathematics if one wants to prove the consistency of classical mathematics, or even that of classical number theory. Consequently, since finitary mathematics is defined as the mathematics in which evidence rests on what is intuitive, certain abstract notions are required for the proof of the consistency of number theory.... In the absence of a precise notion of what it means to be evident, either in the intuitive or in the abstract realm, we have no strict proof of Bernays' assertion; practically speaking, however, there can be no doubt that it is correct...
Paul BernaysRelated topics
absence abstract beyond certain classical consistency correct doubt intuitive mathematics necessary number proof prove sense speaking using meansRelated quotes
From a Darwinian point of view, human beliefs are adaptations to our part of the world. No doubt much of what we believe must be roughly accurate, or else we would not have survived. But the beliefs we have evolved might latch on to the world only enough to help us stumble our way through it, and then only for the time being. Human belief-systems could be useful illusions, appearing and disappearing as they prove to be more or less advantageous in the random walk of natural selection. Might not evolution be one of these illusions? Scientific naturalism is the theory that human beliefs are evolutionary adaptations whose survival has nothing to do with their truth. But in that case scientific naturalism is self-defeating, since on its own premises scientific theories cannot be known to be true.
John N. Gray
Assumptions are dangerous things to make, and like all dangerous things to make--bombs for instance, or strawberry shortcake--if you make even the tiniest mistake you can find yourself in terrible trouble. Making assumptions simply means believing things are a certain way with little or no evidence that shows you are correct, and you can see how this can lead to terrible trouble. For instance, one morning you might wake up and make the assumption that your bed was in the same place that it always was, even though you would have no real evidence that this was so. But when you got out of your bed, you might discover that it has floated out to sea, and now you would be in terrible trouble all because of the incorrect assumption that you'd made. You can see that it is better not to make too many assumptions, particularly in the morning.
Daniel Handler
[A rival theory that has to be better than the dominant "90 percent explanation” cannot go wrong very often and still stay in the contest. For the challenger of a 50 percent incumbent, some shortcomings are not that fatal. This comparative situation has to be remembered as argued by James Burnham:]... we must keep in mind an obvious principle of scientific method. To disprove the theory, it is not enough to show that it is not 100% certain,that difficulties confront it, and certain evidence seems to be against it. It must be further shown that it is less certain than alternative theories covering the same subject matter, that there are in its case more difficulties, more negative evidence than in the case of at least one alternative theory. No theory about what actually happens and will happen is ever ‘certain'.
James Burnham
My work is more varied than at any other point in my life. I am still carrying out research in pure mathematics. And I am working on an idea that I had several years ago on negative dimensions. ... Negative dimensions are a way of measuring how empty something is. In mathematics, only one set is called empty. It contains nothing whatsoever. But I argued that some sets are emptier than others in a certain useful way. It is an idea that almost everyone greets with great suspicion, thinking I've gone soft in the brain in my old age. Then I explain it and people realise it is obvious. Now I'm developing the idea fully with a colleague. I have high hopes that once we write it down properly and give a few lectures about it at suitable places that negative dimensions will become standard in mathematics.
Benoît Mandelbrot
From the first appearance of man upon the earth, down to very recent times, the words "stranger" and "enemy" were quite or almost, synonymous. Long after civilized nations had defined robbery and murder as high crimes, and had affixed severe punishments to them, when practiced among and upon their own people respectively, it was deemed no offence, but even meritorious, to rob, and murder, and enslave strangers, whether as nations or as individuals. Even yet, this has not totally disappeared. The man of the highest moral cultivation, in spite of all which abstract principle can do, likes him whom he does know, much better than him whom he does not know. To correct the evils, great and small, which spring from want of sympathy, and from positive enmity, among strangers, as nations, or as individuals, is one of the highest functions of civilization.
Abraham Lincoln