A few days ago I had an online conversation with Dr Daniel Rubin who is a mathematician living in the US and who works in analysis, geometry and approximation theory. The topic was one close to my heart: Daniel wanted to hear of my objections to the status quo concerning the foundations of modern analysis: namely my rejection of “real number arithmetic” and why I don’t accept “completed infinite processes”. And naturally he wanted to do his best to rebut them.
Here is a link to our chat:
It is certainly encouraging to see that some analysts are willing to engage with the uncomfortable idea that their discipline might actually be in serious logical difficulties. Most of us are reluctant to accept that something we have been working on for years and years might actually be wrong. I applaud Daniel for the courage to engage with these important ideas, and to consider how they fit, or don’t fit, into his current view on analysis.
When we learn pure mathematics, there are many things that we at first don’t understand, perhaps because they are obscure, or perhaps because we are not smart enough — it is easy not to be sure which. Our usual reaction to that is: let me try to accept the things which are cloudy, and hopefully with further learning things will become clearer. This is a reasonable approach to tackling such a difficult subject. However it does require us to put aside our natural skepticism, and accept what the more established figures are telling us at critical points in the theoretical development, even if we imagine this is only temporary.
A good example is: “analysis is built from axiomatic set theory.” In other words the foundations of “infinite sets” and so the basic logical structure of the “arithmetic of real numbers” is a consequence of work of logicians, and can be taken for granted without much further inquiry. Or to put it less politely: it is not the job of an analyst to work out clearly the foundations of the subject; this is something that can be outsourced.
In this fashion dubious logical sleights of hand can creep into an area, transmitted from generation to generation and strengthened with each repeat. Young academics in pure mathematics are under a lot of pressure to publish to obtain a foothold in the academic ladder. This means they do not often have time to mull over those knotty foundational questions that might have been bugging them secretly at the backs of their minds. They probably don’t spend a lot of time on the history of these problems, many of which go back centuries, and in former times engaged the interest of many prominent mathematicians.
Later in their career, if our young PhD has been lucky enough to score an academic job, they might be in a position to go back over these core problems and think them through more carefully. But even then there is often not a lot of “academic reward” in doing so: their fellows are not particularly interested in endeavors that are critical of the orthodoxy — pure mathematics is quite different in this regard than science or even applied mathematics!
And journals are uniformly not keen on publishing papers on foundational issues, especially ones which challenge accepted beliefs. As pure mathematics rests on a premise of logical correctness, any questioning of that is seen as subversive to the entire discipline.
But maybe some serious consideration and debate of the underlying logical structure is just what the discipline really needs.
I certainly enjoyed our conversation and I think there are valuable points in it. I hope you enjoy it, and look forward to another public YouTube discussion with Daniel.