We usually conceive of space-time as a continuum. Or we think we do. We certainly experience it as a continuum. We usually see the number line as a working model of this continuum. With this model we can create an infinitely-divisible co-ordinate system and then map all locations in space-time onto the numbers. It may even be difficult to see it any other way.
On analysis, however, it is found that the worldview arising from this combination of views is paradoxical. This becomes clear when we examine the foundations of analysis. The problem is discussed at length by Hermann Weyl in Das Continuum, a famous book on the foundations of analysis. It is a well-known problem in mathematics, albeit that it would often be seen not as a problem for mathematics but, rather, metaphysics. Either way, mathematics gives us an appropriate language with which to talk about it.
A study of the continuum problem would be a quick way of establishing that there is something fundamentally wrong with the idea that space-time is extended in the same way as the number line, and thus something wrong with the stereotypically ‘Western’ (and still essentially Newtonian) worldview. This problem is foundational for any tradition of thought that rejects mysticism. For the philosophy of the East, however, a difficulty would arise only if this fatal crack failed to appear in the foundations of the typically Western metaphysical worldview. It would predict such a flaw, always has and always will, and would be implausible if there were no evidence of it.
It is all a bit mind-bending. A continuum cannot have parts and so it cannot be extended. To be a continuum it would have to be unmanifest. The arithmetical line can be manifest, and is clearly a perfectly useful tool for navigation around the co-ordinate system we call space-time. Yet it cannot be the case that extended space-time has the same properties as the number line. It would not make sense.
In an article examining this problem recently submitted to the modest online journal Philosophy Pathways, which was readily accepted at the last minute because I was guest editor this month, I quote at some length from the writings of Hermann Weyl, Tobias Dantzig and John Bell on this topic. I find it a fascinating problem and believe it to be extremely important in philosophy and much too often overlooked.
Anyway, this is an advertisement for the article. It has the same title as this post and is listed under pages here…
The World Knot 