The nine questions of the 13^th part of this blog are, of course, unanswerable. The notion that something described as a ‘transfinite set’ can exist is quite clearly a chimera. Cantor’s Diagonal Argument and its generalised corollary indicate that these totalities necessarily have the potential to grow unpredictably, and therefore they are not ‘sets’, nor in the usual sense ‘mathematical objects’.  A set as a mathematical object must (a) consist only of elements which are mathematical objects and (b) it must, as a totality, be settled, otherwise it is not a timeless mathematical object. To postulate ghostly, unimaginable objects simply to “save” Cantor’s invalid theory is ridiculous.

     More generally, the notion that we can still sensibly regard mathematics as being about an ‘Abstract World’ different from the Physical World —but equally autonomous and real— is quaint.  Some of today’s mathematic gurus seem to have absent-mindedly nailed their colours to a Medieval mast which countenances metaphysics —trick thinking which has no credibility today.  Such gurus claim that the experience of doing pure mathematics has convinced them that it is “there”  —a form of otherness which can’t be talked away. But what they are observing is that it is timeless, so that it doesn’t change when you go back to it after a gap of days or months. Of course it is: it is we who insist on this timelessness when we introduce new definitions into mathematics.  Our predecessors insisted on this timelessness when they introduced definitions. Mathematics is an edifice built over many centuries by exceptional individuals who insisted on rigour and consistency.  Until fairly recently it was regarded as the ‘Heartland of Truth’.  This is a reflection of the high standards and determination of its constructors, not a sign that it was handed down by the Gods, or that it has somehow reified itself into existence from a shadowy “Other Reality”.

   Let’s consider a single tally /.The tally you

 inscribed in your notebook last month will still be there when you go back to it next week. This doesn’t mean that there is some crystalline, abstract “real stuff” out there (metaphysically supporting it?) —only that tallies are semi-permanent, humanly inscribed, marks on paper.

So what is the hidden roadblock which is preventing a recognition that metaphysics and transfinite totalities are nonsense?

I’m afraid the answer is Russell’s legacy. At the time when he was most active, Russell was, probably, the most highly regarded figure among the gurus of the 1900s. Among the ordinary Anglophone public he was probably, over his lifetime, the most influential philosopher of mathematics. The Oxford Companion to Philosophy describes the book he wrote with help from Whitehead as being (sic) <<considered by many to be the one of the great intellectual achievements of all time>> (p. 909). That Russell still receives this kind of preposterous adulation today is a sign of how little such commentators have understood the grievous flaws in his legacy.

Many Anglophone commentators are still uncritically swallowing Russell’s claim that Principia Mathematica “proved that mathematics can be reduced to logic”. They seem to be quite unaware (1) that Russell was only able to “complete” this so-called ‘logical reduction’ by co-opting three unobvious axioms, the axioms of infinity, choice and reducibility. [These were supposed axioms which he discovered that he needed —if he was to get his “reduction” to work.  To pluck these out of the blue and invest them with apparent authority is tantamount to the schoolboy ploy which teachers jump on, when supposed axioms or principles are invented to justify fallacious arguments.] 

They are also unaware (2) that he invented a wholly unobvious principle, the notion that meaning comes on ‘levels’ or ‘stratifications’. They (the Anglophone public) also seem to be quite unaware that what Russell  claimed  to have done implied —if we treat words in the most natural way— that the logic involved was self-evident.  But Russell’s logic (3) was miles away from being self-evident, because he needed these four unself-evident assumptions to get it to work. Without these unobvious assumptions it didn’t work.

Let’s look at each of these pseudo-axioms in turn.

The pseudo-axiom that an infinity of objects exist is pure metaphysics. It flatly defies the common perception that we never see, feel or hear-of an infinite collection of objects in the real world.

The pseudo-axiom of choice asserts that an infinite selection of arbitrary (undefined) choices of an item from each of an infinity of pots exists. This flatly assumes that we can claim to have “got” a specific selection of an infinity of such items when we haven’t got a formula or rule of any kind to fix them.  Here again it is asserting that something “exists” which doesn’t. 

The pseudo-axiom of reducibility is inexcusably ad hoc.  There is no reason whatever to think that truths about objects on the upper levels of Russell’s imagined stratification coincide with the corresponding truths about objects on the lower levels.

Stratification is an imposition of arbitrary structure onto logic, contrary to commonsense. Russell analysed the paradox of the Liar by saying that the Liar “should” have remarked that he was making a statement on level n.  This was a statement about a statement on level n, so it counted as level n+1. Thus Russell has tricked the Liar into making a claim which is palpably false (i.e. he is not making a statement of level n).  This beggars belief as a serious way to try to explain how the Liar’s statement fails to make sense. Russell has no magic wand to force the stratification he has dreamt-up onto the Liar. So Russell’s ‘should’ has no logical power.

Thus Principia Mathematica (PM) certainly does not show that mathematics can be reduced to logic in the normal sense of ‘logic’ (self-evident logic). Logic is, or should be, the science of clear thinking. It has no room for pseudo-axioms. But PM does not even do this on its own terms, because the kind of ‘logic’ Russell is reducing it to is mathematical logic.

There is also a much more obvious, visible absurdity about Russell’s Principia Mathematica project as a whole.  He is, officially, engaged in a plan to reduce ordinary mathematics to mathematical logic. But this so called ‘mathematical logic’ is just another kind of more abstruse mathematics.  The notion that such a reduction —even if it were correctly accomplished— would explain anything about mathematics is absurd.  Almost no one is “more at home in” mathematical logic than in mathematics.  The idea that it is more self-evident is ridiculous. Russell spent more than 80 pages proving that 1+1 equals 2. It is the most ludicrous waste of time ever presented as logical enlightenment. (There is no enlightenment involved: it is a process which substitutes something obscure for what was, anyway, fairly obvious.)

Russell made great play with the claim that he had discovered proper definitions for ‘the number 2’, ‘the number six’, etc.  But this is just scholasticism; this so-called ‘problem’ is a non-problem. 

Russell was operating in the era before the linguistic enlightenment introduced by Wittgenstein. He (Russell) was a Platonist who still treated the meaning a word as what it named. But meaning, we now know, is much more than mere naming. (It is quite obvious that words such as ‘but’ and ‘of’ do not name anything.) The definition of the numeral 2 is 2=1+1, for the numeral 6 it is 1+1+1+1+1+1.  These are simply respectable ways of saying that 2 is shorthand for // and 6 is shorthand for //////.

‘The number six’ is simply a reification of the numeral 6 which kept popping up in the calculations of the arithmeticians of Antiquity. Using this reification adds gravitas to the profession of arithmetician, much as the introduction of the idiom <<The marriage of A and B>> adds weight, social presence and gravitas to an intimate bond between two people, A and B.

CHRISTOPHER ORMELL July 1^st 2021.