The fanciful notion that mathematics is, broadly speaking, an inquiry into a world of ‘Real Abstract Objects’ quite independent of humanity —and just as real as physical things— has often been cited by mathematicians of the highest calibre as the main justification for their lifelong commitment to it.  These Abstract Mountains are there, they assert, and the fact that they are “there” is a very good reason for trying to climb them.

But there is a nettle here which we need to grasp. Could this be another example of unsound thinking about the wider implications of mathematics? The four howling mistakes named in Blog 16, one might think, could never happen. But the truth is that they did happen, and sceptical second-thoughts about them at the time were brushed aside. We have to face the fact that many outstanding mathematicians chose their career, because they were attracted by the aesthetic perfection of mathematics, and they preferred to operate in this pleasingly ordered world rather than in the messy, tricky, complicated, ambiguous landscapes of ordinary reality. If there is any moral to be drawn from this sorry story, if can only be that we all need to exercise circumspection before accepting supposedly “obvious” way-out in-house generalisations about maths.

The historical evidence is totally against any metaphysical narrative like “Real Abstract Objects”. At all points in the history of mathematics, mathematicians have explored possible new definitions, searching for those which sit well with our existing knowledge. Imre Lakatos showed vividly in his classic series of articles Proofs and Refutations (1960-62) that these explorations were invariably similar to those undertaken by scientists when they tried to explain infuriatingly puzzling phenomena. There were muddles, mistakes, partial insights, back-trackings and ambiguities to be overcome. Afterwards the whole saga was presented, in retrospect, as a smooth, swift, royal road to crystalline abstract forms.  The struggle-to-understand was airbrushed out, and the final results were streamlined and refined… in a way considered appropriate for such an august, Olympian subject as mathematics. 

Unfortunately, this well-meaning approach creates a false impression.   

The actual story of mathematics uncannily resembles that of a self-made man, who was born in a primitive hovel, but managed somehow to struggle out of poverty. He learnt to read and write, and gradually improved himself… until he became rich, famous, and a member of the House of Lords. This is what the record tells us at every point.  The impression given to students in standard textbooks is, by comparison, an ahistorical PR construction —one generally thought to be glorious, and fitting for such a high-level, privileged super-subject. (The low level of regard for maths today among the general public is quite absent within the subject.)

Two mathematicians remark candidly in a current publication for schools:

<<How is it that mathematics which seems to be a construct of the human mind can, nevertheless, be applied to the real world with astonishing results?…That mathematics is so successful in the real world remains an enigma.>>

[Benjamin Baumslag and Jack Jacobson in Mathematics in School 2021]

This excerpt shows the power of the kind of misapprehension which the application of glorifying PR produces. Even though the two mathematicians have not swallowed the Real Abstract Mountains line, they have become oblivious of the long struggles involved in perfecting instrumental mathematics. There is no problem about the applicative power of mathematics —if we view it realistically from an honest historical point of view. Proto-mathematics using tally-bundles to represent groups of things like flocks of sheep or hauls of fish, was used for thousands of years before the idea of defining numbers was probably even considered. These early tally bundles were used to “keep tabs on” specially important collections of things which might go missing. They established records, which told the shepherd that some of his sheep had got lost, or the fisherman that some of his fish had been stolen. 

In the ancient world mathematics (in the form of logistics) was used by Kings, Emperors, Generals etc. to plan their military campaigns efficiently, often making the difference between victory or defeat. They also used geometry to plan handsome buildings, triumphant arches, tombs, etc. In all these cases mathematics was being treated in a way we nowadays call ‘mathematical modelling’. Working with putative numbers meant that potential shortages of weapons, horses, chariots, etc. could be foreseen —and subsequently avoided. This is how mathematics has been growing its effectiveness in the real world. It provides a vital source of foresight in the build-up towards any practical project. From the 17^th century onwards it provided a way of finding viable new theories in science —from unviable ones.  A putative theory, expressed by a set of equations, could be investigated mathematically, by looking to see what they predicted would happen. If this coincided with what actually happened, the theory was viable. If not, it wasn’t.

Thus the secret which explains the so-called ‘enigma’ raised by Baumslag and Jacobson is that mathematics has been highly valued all through history by monarchs, generals, architects, scientists… because mathematical modelling provides priceless vision about the prospects for plans, projects, schemes, theories… innovations of all kinds, provided that they were based on a skeleton of predictable elements.

This is Charles Peirce’s much neglected definition of mathematics as <<The science of hypothesis>>.  Any proposed innovation, whether practical or theoretical, is, in effect, an hypothesis —to the effect that what is proposed will meet some new needs or fit some new puzzling facts.

In the process of developing mathematical modelling all kinds of new functions, procedures, etc have been devised (defined). The total body of mathematics grows every time a new definition is generally accepted as being a useful addition to the repertoire of the discipline.  Until the beginning of modern mathematics (around 1830) virtually every definition in mathematics was devised to increase its problem-solving power in the real world. After 1830 it became possible to create new mathematics, consciously, out of the blue. This capacity to create such a wonderful thing as mathematics has, unfortunately, been too intoxicating for the subject’s own good. 

CHRISTOPHER ORMELL September 1^st 2021.