In this instalment we look at the most howling mistake ever made in maths. But it went unnoticed.  It was then, unaccountably, accepted into the official body of mathematical thinking. This is a howler which has generated untold consequences. It was a very simple mistake,  but subsequent scholars evidently took it on trust, and eventually it became unquestioned dogma.

In the mid 19^th century George Boole discovered set theory. He was a largely self-taught mathematician, a bit out of the mainstream. His theory of sets was quite original. It featured a new form of algebra somewhat like the abstract algebra of Galois, though of a different kind. So it seems to have been firmly treated as “abstract algebra” from the beginning.  Boole used A+B for the union of two sets A and B (AUB) and  AB for the intersection of A and B A∩B.

This looked very like yet another example of new, strange, non-standard abstract algebra, as pioneered by Galois. It was unequivocally pure, abstract mathematics, wasn’t it? Like Galois Theory, it had not arisen from abstractions culled from empirical research… So, perhaps unconsciously relying on the comparison of Boole’s new algebra with Galois’, everybody seemed to assume from the beginning that Boolean theory was also pure mathematics. 

Whether Boole himself thought of it in this way is another matter. He called his book The Laws of Thought, thus treating it as if it were a kind of meta-psychology.

At the time the mathematics community was busy digesting the new abstractions of complex theory and non-Euclidean geometry, so the common focus had moved a long way away from applications.  There were some physicists doing applied mathematics, but virtually the whole body of professional mathematicians were doing pure mathematics.  Incidentally the coordinate geometry of Descartes effectively removed the need for the kind of sustained creativity and reasoning which had been necessary to solve geometrical problems by the use of Euclidean methods. This seems to have had an effect on the workstyle of pure mathematicians, inclining them to pursue powerful generalising concepts and to eschew the use of imaginative curiosity.     

No one seems to have asked the simple question: <<if Boolean Algebra is “pure mathematics”, what is the application of this “pure mathematics” going to look like?>>.

This question can only non-plus anyone who asks it, because Boolean Algebra is already about sets of things, some of which are physical realities, or things like the days of the week which are experienced in a similar way. This is already —plainly— “applied mathematics”.

Of course geometry began with a similar ambiguity, namely that the lines and points of geometry seemed to be referring to lines and points marked on papyrus or scratched on clay. But Plato saw, correctly, that this was a mistake: the lines and points studied in geometry are not real lines and points. They are, rather, Abstract Objects in an Ideal Space, not physical objects in physical space. 

Descartes’ coordinate geometry, as mentioned earlier, transformed geometry at a stroke into arithmetic and generalised arithmetic (algebra). So the norm of post-Cartesian mathematics is that all mathematical objects are to be regarded as abstract objects, quite distinct from real objects. Geometric truths are essentially truths about numbers. Godel’s later arithmetisation of mathematical logic incidentally showed that modern abstract algebra (such as Galois’) can also be brought under this umbrella. Mathematics consists of a priori truths which owe nothing to empirical observation or experiment.

There are truths in mathematics which we appreciate because we trust them completely, just as there are truths in chess. But neither the truths of mathematics nor the truths of chess  are “truths about the real world”.

So set theory got off to a rather muddled start. It was obvious, really, that “the set of dinner plates in Mansion House”, which can no doubt be divided into various sub-sets, is not a mathematical object of any kind.    This is obviously “set theory applied to the real world”.

So pure set theory can only deal with sets of mathematical objects, such as the set of integers which are equi-distant from adjacent prime numbers {6,9,15,21,26…}.

A set is a collection of objects, and a set in pure mathematics can only be a set of mathematical objects. 

A major conclusion follows from this observation: sets cannot be the fundamental objects of mathematics. Because sets in mathematics are collections of mathematical objects they can only be defined after we already know what counts as a ‘mathematical object’.  It follows that the entire project of “re-building mathematics on a set basis”, which was for many years an Holy Grail — is a conceptual mistake. It was pushed with much zeal by Dedekind, Cantor, Frege, Russell, Whitehead, The Bourbaki and “New Maths for Schools”… An extreme example occurred when John Conway tried to build the whole of mathematics using the null set as a universal building block. All this much vaunted effort was misplaced. Trying to re-build mathematics using sets has never been successfully accomplished. This should not surprise us because the project was based on a logical mistake.

It is quite ironic that a poorly-focused development, set theory, should have become the favoured way, for more than a century, to try to restore rigour in mathematics. Rigour in mathematics has to encompass the meaning of words of ordinary language like ‘truth’, ‘is a member of’, ‘is always’, ‘is a collection’… so it must incorporate, inter alia, a coherent account of meaning in ordinary language.   This is what the Official Story has systematically overlooked. John Lucas, who died earlier this year, formulated the bombshell insight arising from a study of ordinary language: that to say that something is true is to say that it can be trusted.