In the cited chapter of Science and Hypothesis, Poincare was trying to show how the physical continuum (numbers in the physical world) influences the construction of the mathematical continuum (real numbers in mathematics), but that there are some abstractions such as exact equality that only make sense in the latter.
>From the Preface, summarizing the passage under discussion: ...we have to analyse another fundamental idea, that of mathematical magnitude. Do we find it in nature, or have we ourseleves introduced it? And if the latter be the case, are we not running a risk of coming to incorrect conclusions all round? Comparing the rough data of our senses with that extremely complex and subtle conception which mathematicians call magnitude, we are compelled to recognise a divergence. The framework into which we wish to make everything fit is one of our own construction; but we did not construct it at random, we constructed it by measurement so to speak; and that is why we can fit the facts into it without altering their essential qualities. Best wishes, John Miller, Raul D wrote: > Sylvain Baron wrote: >> In 1902, in a book called "Science and Hypothesis" Henri Poincare >> wrote about physical and mathematical continuum : > " We cannot believe that two quantities which are equal to a third are > not > " equal to one another > > Is this an axiom? > > Trivial counter-examples are easy to find (though, granted, > it could be said that they do not constitute examples of > continuum). > > For example, if I have 3 dollars, and you have 3 dollars, > you could say that these two quantities are equal. But the > same object can't exist in two places at the same time, so > either: > > (a) the three dollars that I have are different from the > three dollars that you have, or > > (b) we don't really each have three dollars. > > If two quantities which are equal really represent something > different from each other, why should that quality cease to > exist when a third quantity is introduced? > > Put differently, as long as the concepts exist purely in one's > mind, equality is easy to manage. However, in the real world > there is no such thing as absolute equality. Even the same > object is different from two different perspectives. This is > not simply a matter of appearance, and is not simply because it > is changing over time. But, also because the distinction between > what is a part of the object and what is not is an artifact of > the mind. For example, is that photon or that oxygen molecule > or whatever other small thing near the surface of the object a > part of it or not? > > Granted, concepts of equality are incredibly useful tools of > thought. > > But I think Henri Poincare talked about this as an issue of > belief for a very good reason. > > -- > Raul > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
