Against the anti-platonists

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (WMCF) by Lakoff and Núñez, is an attempt to explain how we understand mathematics, using principles developped in cognitive theory. This is a perfectly worthy and reasonable goal. They do however not stop there, but try to use their ideas to argue that mathematical platonism is impossible, a romantical mistake.
Let's take a look at a thing they say in the book (paragraph below is copied from Wikipedia):

WMCF (p. 151) includes the following example of what the authors term "metaphorical ambiguity." Take the set A={{},{,{}}}. Then recall two bits of standard elementary set theory:
The recursive construction of the ordinal natural numbers, whereby 0 is , and n is n-1 {n-1}.
The ordered pair (a,b), defined as {{a},{a,b}}.
By (1), A is the set {1,2}. But (1) and (2) together say that A is also the ordered pair (0,1). Both statements cannot be correct; the ordered pair (0,1) and the unordered pair {1,2} are fully distinct concepts. Lakoff and Johnson (1999) term this situation "metaphorically ambiguous." This simple example calls into question any Platonistic foundations for mathematics.

There are several clear errors here. The claim that two statements cannot both be correct because they refer to "fully distinct concepts" is not only a vague statement with no backing, it also assumes that the two instances above of the number 2 have nothing to do with each other. This is manifestly not the case, since going through the simple definitions above give a direct way of passing from one instance to the other. As an example, the fact that the two expressions
e^{\pi i} and -1 seem to be "fully distinct concepts", does not prevent them from being equal as real numbers.
The "metaphorically ambiguous" situations are precisely cases where we make connections not through intuition or metaphor, but by logic. This shows that part of mathematics consists of establishing truths which are not obviously grounded in anything in our experience as embodied beings, but rather emerge from the inner logical structure of mathematics itself. These situations are therefore examples whose existence Lakoff's and Núñez's theory fails to explain.
Therefore, rather than calling Platonism into question, these examples only serve to affirm the fact that analogy based in physical reality does not account for all aspects of mathematics.
Such careless arguments against platonism shows that Lakoff and Núñez, rather than being unbiased scientists searching for the truth whatever it may be, are on a crusade to defeat competing philosophies in order to promote their own. Mathematicians know that while these ideas are very interesting for mathematical education and in understanding how we think about mathematics, it basically says nothing about the ontological status of mathematics itself. It can at best support a viewpoint which takes as its basic principle the human understanding of mathematics, but it can never determine what the nature of mathematical objects is. This makes the theory completely consistent with platonism.

Of course this is all part of the contemporary trend in philosophy to reject platonism, dualism, and whatever philosophical theory there might be which has had a great influence for a long time. The more support a classical theory has had through the ages, the bigger the victory for those who manage to dethrone it. History has shown that such ambitions usually do not lead us closer to understanding and consensus about the world.