How is it that a person trained in the application of logic can continue to ignore the contradiction that lies at the heart of conventional set theory - that one limitlessly large quantity can be smaller than another limitlessly large quantity? (Footnote: For more on this, see Proof of more Real numbers than Natural numbers, Cardinal Numbers and Cantor's diagonal proof.)
There is a commonly accepted mathematical proof principle to the effect that if one can demonstrate that an argument which requires certain assumptions leads to a contradiction, then one or more of those assumptions is untenable or there is one or more illogical steps in the argument. But mathematicians have turned this principle on its head by assuming that if two infinite sets have no one-to-one correspondence, then if they were the same size, that would be a contradiction - and they proceed to “solve” this supposed problem by replacing it with a blatant contradiction - that one limitlessly large quantity can be smaller than another limitlessly large quantity.
It would appear that the initial presentation of the argument to a student (usually based on some version of Cantor's diagonal proof) is as an argument that starts off with a selected set of assumptions and ends in an outcome which entails the contradiction that one limitlessly large quantity can be smaller than another limitlessly large quantity. At that point, the student has the choice of:
- Accepting the conventional argument, and simply turn a blind eye to the contradiction, or
- Accepting the conventional argument, but continue to try to unearth the cause of the contradiction, or
- Rejecting the conventional argument, and continue to try to unearth the cause of the contradiction, or
- Rejecting the conventional argument, without giving it any further thought or investigation.
For someone planning a career in mathematics or logic, the last option would not appear to be a likely choice, while the third option could be seen as tantamount to career suicide, with expectations that any career progression would be minuscule.
For the careerist minded, the first or second options would appear to be the only sensible choice. However, it appears to be the case that very few choose the second option - or else they keep their thoughts on the matter only to themselves and do not share any of their concerns about the contradiction with colleagues. It is certainly the case that one has great difficulty in finding any academic who admits to any unease about the contradiction; one might mention Alexander Zenkin and Ludwig Wittgenstein but they are both deceased, leaving Wolfgang Mückenheim and Norman Wildberger (Norman Wildberger on set theory (PDF)) as surviving living academics who disagree with the conventional stance.
It appears that the vast majority slip into accordance with the current philosophy where there is an implicit agreed convention to turn a blind eye to the contradiction, and continue the pretense that it does not exist. Of course, some students may attempt to follow one of the other options for a while before realizing that their future career is bleak if they persevere in that vein, and sooner or later they bow to convention.
In this way, a body of mathematicians and logicians has become established which has been inculcated with an inbuilt resistance to any suggestion that there might actually be a simple logically valid way of removing the contradiction. The fact that there has been no devious intention behind this change does not alter the fact that it has occurred. The passive acceptance of an inherent contradiction was not arrived at by purely logical considerations; that acceptance was an acceptance of a purely ad hoc declaration which has now become dogma. Daniel Kahnemann notes that: (Footnote: Daniel Kahneman, Thinking, fast and slow, Macmillan, 2011.)
“…the illusions of validity and skill are supported by a powerful professional culture. We know that people can maintain an unshakable faith in any proposition, however absurd, when they are sustained by a community of like-minded believers. Given the professional culture…, it is not surprising that large numbers of individuals in that world believe themselves to be among the chosen few who can do what they believe others cannot.”
While Kahnemann writes this in relation to the faith of financial forecasters in their pronouncements, it may be that the same feeling of security in belonging to a group helps to sustain the belief in mathematical absurdities.
See Why Facts Don’t Change Our Minds (by Elizabeth Kolbert of The New Yorker), an article that looks at why there can be a remarkable persistence of belief in the face of contradictory evidence. The initial inculcation may be the reason why it is so difficult to engage a mathematician/logician in any logical discussion about the contradiction. In the same way as a bible thumping fundamentalist adheres to his ingrained beliefs that have no logical basis, the mathematician/logician has subconsciously allowed the dogma that the contradiction is to be ignored to be lodged deep into his mindset. He will deny any such dogma, and instead will plead that there is no contradiction involved, and that the other person who is claiming that there is a contradiction does not understand the subtleties of the situation. There is a striking analogy here with a religious fanatic claiming that the unbeliever does not understand the subtleties of his god's intentions.
But there is a real contradiction. By definition, an infinitely large quantity of things is a limitlessly large quantity of things, and so there is no limit on that quantity, which means that it cannot possibly be a smaller quantity than any other quantity. It is simply absurd to pretend that there is no contradiction.
So why do mathematicians/logicians continue to evade this contradiction? Presumably a large part of the problem is that intelligent people are good at creating arguments to support a given position; however, the fact that they are intelligent does not necessarily mean that their arguments are completely logically watertight. (Footnote: See chapter 18 of the book Why people believe weird things by Michael Shermer, Macmillan, 2002.) If you actually examine such arguments, you will find that they are logically invalid or that they include at least one of several common argumentative fallacies such as a deflection from the subject in question. (Footnote: See, for example, Dishonest tricks used in Arguments, A List of Fallacious Arguments, Fallacies - The Internet Encyclopedia of Philosophy, Logical Fallacies Handlist, Wikipedia entry - List of Fallacies.) As an example of how a mathematician who surely can be considered very intelligent can try to defend the indefensible see A typical defence of different sizes of infinity.
The mathematician/logician is stuck with a mindset that believes that there cannot be anything further to be said about the original argument. He refuses to accept that a complete logical meta-linguistic analysis of the argument can give a complete resolution of the contradiction. Part of the problem may be that if a person has a prior belief that the problem can only be resolved by mathematical symbol manipulation, or that there actually is a rigorous proof that one limitlessly large quantity can be smaller than another limitlessly large quantity, (there isn’t(Footnote: See Proof of more Real numbers than Natural numbers)) then they may feel justified in summarily rejecting any proposed alternative. And it seems to be the case that this is precisely what mathematicians do.
Of course there is another factor that makes some mathematicians unwilling to even consider that they might be wrong. That factor is that accepting the solution that resolves the contradiction also means that they must also reject a large part of their subject of study - the study of ‘limitlessly large sets’ that have ‘different sizes’ have constituted the major part of the life’s work of many mathematicians. One can understand the reluctance of such people to contemplate any thought of a major reconsideration of the foundations of their area of study - but at the same time that cannot excuse the subjugation of logic so that it has become subservient to convention and convenience.
Beauty in mathematics
A theme frequently invoked by mathematicians is that mathematics, especially those parts that are considered to be pure mathematics, has an innate beauty. For example, G Hardy wrote:
“The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.” (Footnote: G H Hardy, A Mathematician’s Apology, Electronic Edition, 2005)
This notion of inherent beauty sits very uncomfortably with a mathematics that requires one to accept a blatant contradiction. How can a mathematics that results in such an ugly contradictory blemish ever be considered beautiful? But, amazingly, instead of finding it ugly, mathematicians consider it to be a paradise - as David Hilbert remarked:
“No one will drive us from the paradise which Cantor created for us.” (Footnote: D Hilbert, Über das Unendliche, Mathematische Annalen, 95, pp. 161–190, 1926, Original German text online.)
How can anyone reconcile the notion of a paradise based on a lie with a profound desire for ultimate beauty? To which the answer might be: in the same way that religions preach visions of paradise - visions based entirely on irrational belief and which have no logical justification.
For further reading on why people continue to believe things in the face of contrary evidence, see:
Denying to the grave: Why we ignore the facts that will save us by Sara and Jack Gorman, Oxford University Press, 2016.
The Enigma of Reason by Hugo Mercier and Dan Sperber, Harvard University Press, 2017.
The knowledge illusion: Why we never think alone by Steven Sloman and Philip Fernbach, Penguin, 2018.
Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked. Difficulties in understanding the site content are usually best addressed by contacting me by e-mail. Note: you will be asked to provide an e-mail address - any address will do, it does not require verification. Your e-mail will only be used to notify you of replies to your comments - it will never be used for any other purpose and will not be displayed. If you cannot see any comments below, see Why isn’t the comment box loading?.