... mathematics is the basis of any reality, conceivable or inconceivable, as it is the basis for physical reality ...

Rafael Núñez and George Lakoff (of Lakoff And Johnson) oppose this notion, which they see as one facet of the "Romance of Mathematics". Indeed this is the core article of faith of the mathematical romantic.

The tenets of mathematical romance are, according to Lakoff and Núñez, that:

• mathematics is objective, absolute and certain
• mathematics would be the the same with or without human presence in the universe
• mathematicians discover truths that apply throughout this and every other possible universe
• mathematical logic defines rationality...
• ...therefore mathematics is the highest form of human intelletual achievement
• the mathematics of physics is somehow in the phenomena studied
• only mathematics can describe mathematics

It's not neccesary to believe all of them to be a mathematical romantic.

They claim, in their long and complex book Where Mathematics Comes From (ISBN 0465037712), that these notions are not just wrong but also damaging. Instead, they offer their theory of the embodied mind

This theory begins with the observation that humans (and other animals, but especially humans) have a capacity called subitizing, the ability to immediately, that is, without conscious thought, distinguish between collections of one, two, three or four objects. Four is a typical upper limit.

From this innate ability, they argue, we gain out concept of number (cardinality, in fact). Then, for example, by using the metaphor Arithmetic is Object Collection, the notion of an empty collection is pondered, and so the concept of zero (the cardinality of an empty collection) arises. Zero is not a subitized number, so is not built into us, hence its late arrival on the mathematical scene.

And so on.

The book is a beast. As with others of Lakoff's works (singly or in collaboration) one doesn't have to accept, or even understand, all his arguments to find it intersting and stimulating. I've read only the indicated highlights of the book, and been fascinated, challenged and puzzled in equal measure so far. That's what I call value for money.

Interestingly, while they reject the beliefs about mathematics outlined above, they are adamant that the embodied mind theory is neither relativist (where the romance of mathematics is "absolutist"), nor postmodern (where the romance of mathematics is profoundly modernist).

Trying to think about Lakoff's and Núñez's theory and Gregory Chaitin's at the same time casues the kind of itching in my brain that usually suggests that someone is on to something. -- Keith Braithwaite

Does this beget the Romance Of Quantum Mechanics? -- C Herger Thomann

Lakoff's and Núñez's theory that a human (embodied) mind can only access certain mathematical notions (e.g. the notion of infinity) by means of a metaphor, be it right or wrong, does not contradict the Platonist view of mathematics as existing independently of its understanding by any thinking being. This is a different thing altogether.

Lakoff and Núñez attribute to a mathematical romantic the idea that human mathematics is just a part of abstract, transcendent mathematics. Yet this is not what a Platonist would say. Human knowledge of mathematics is neither abstract nor transcendent, and may well be dependent on the innate ability of subitizing and developed by means of metaphorical thinking. But whether this process leads to creating new ideas or to discovering ones already existing, cannot be determined by studying the process itself. Thus identifying mathematical notions possessed by human minds with their counterparts in the Platonic realm of ideas is not a corollary to the embodied mind theory, but its unwarranted assumption. -- Kasia Dymara