In the most recent edition of Language, Truth and Logic , Professor A. J. Ayer still maintains that pure mathematics is analytic, being in fact merely a vast system of tautology. He is much more confident about this than are most contemporary professional mathematicians who have investigated the foundations of their subject. Following the breakdown of the efforts both of Frege and of Russell and Whitehead to derive pure mathematics from logic, i.e. to prove that the denial of any one (...) proposition of mathematics would necessarily be self-contradictory, Hilbert attempted to prove the more modest thesis that pure mathematics is consistent, i.e. that no two propositions of mathematics can contradict each other; but in 1931 Gödel discovered that even this thesis was undecidable according to the “rules of the game.” As Weyl has recently lamented, “From this history one thing should be clear: we are less certain than ever about the ultimate foundations of mathematics.” The sense in which Ayer uses the terms analytic and tautology implies also that in his view the activities of pure mathematicians lead to nothing new. It is true that he remarks that “there is a sense in which analytic propositions do give us new knowledge. They call attention to linguistic usages of which we might otherwise not be conscious, and they reveal unsuspected implications in our assertions and beliefs.” But, he continues, “we can also see that there is a sense in which they may be said to add nothing to our knowledge. For they may be said to tell us what we know already. ” This denial of novelty in mathematics is as typical of contemporary positivism as the prophecy of Comte that the composition of the stars would never be revealed to us and the objections of Mach to the atomic hypothesis were characteristic of nineteenth century positivism. Indeed, one wonders why the term positivism should have been appropriated by successive philosophers whose common outlook could be so much more fittingly described as negativism. (shrink)

“ ‘Tell me, Protagoras,’ he said, ‘does a single grain of millet or the ten-thousandth part of a grain make any sound when it falls?’ And when Protagoras said it did not, ‘Then,’ asked Zeno, ‘does a bushel of millet make any sound when it falls or not?’ Protagoras answered that it did, whereupon Zeno replied, ‘But surely there is some ratio between a bushel of millet and a single grain or even the ten-thousandth part of a grain'; and when (...) this was admitted, ‘But then surely,’ Zeno said, ‘the ratios of the corresponding sounds to each other will be the same: for as the bodies which make the sounds are to one another, so will the sounds be to one another. And if this is so, and if the bushel of millet makes a sound, then the single grain of millet and the ten-thousandth part of a grain will make a sound.’ This was the way Zeno used to put his questions”. (shrink)

The history of Natural Philosophy is dominated by a paradox; broadly speaking, a vast increase in its range of application to the external world has been accompanied by a sweeping simplification in its basic assumptions. From the standpoint of Empiricism this dual development appears utterly mysterious. On the other hand, Rationalism, which seeks to demonstrate the metaphysical necessity of natural law, and hence might throw light on this development, has been generally discredited, particularly by men of science. It is not (...) surprising, therefore, that philosophical discussion of scientific method has become a Babel of confusing tongues. (shrink)

And the first step of the Peripatetick argument is that, where Aristotle proveth the integrity and perfection of the World, telling us, that it is not a simple line, nor a bare superficies, but a body adorned with Longitude, Latitude and Profundity; and because there are no more dimensions but these three; the World having them, hath all, and having all, is to be concluded perfect. And again, that by simple length, that magnitude is constituted, which is called a line, (...) to which adding breadth, there is formed a Superficies, and yet further adding the altitude or profundity, there results the Body, and after these three the dimensions there is no passing farther, so that in these three the integrity, and to so speak, totality is terminated, which I might but with justice have required Aristotle to have proved to me by necessary consequences, the rather in regard he was able to do it very plainly and speedily. *Received 9. iii. 54. (shrink)

linear time is inadequate to account for all events in our world. In particular, the idea that time may have more than one dimension has been invoked by J. W. Dunne, in his well-known book An Experiment with Time, to justify his claim that ...

It is impossible for us to understand the character of Newton or the state of the physical sciences in the seventeenth century, without referring at length to the life and work of Robert Hooke.’ So writes the author of a recent biography of Newton. Well might we ask with him ‘What manner of man was he whose personal opposition delayed the publication of Newton's Optics for thirty years and almost prevented the completion of the Principia; whose bitter tongue confirmed Newton's (...) tendency to secrecy and isolation, destroyed his early enthusiasm for the Royal Society, and disgusted him with science?’ It is not surprising that a man, capable of all this, should play a conspicuous rôle in the history of natural philosophy. Indeed a study of Hooke's career throws much light on the origins of modern science and, in particular, it helps to determine the much disputed status of mathematics in the development of physical theory. (shrink)