Editing Where Mathematics Comes From (section)

== Critical response ==
{{multiple issues|section=y|{{Original research section|date=March 2010}}
{{More citations needed section|date=July 2007}}}}

Many working mathematicians resist the approach and conclusions of Lakoff and Núñez. Reviews of ''WMCF'' by mathematicians in professional journals, while often respectful of its focus on conceptual strategies and metaphors as paths for understanding mathematics, have taken exception to some of the ''WMCF''{{'}}s philosophical arguments on the grounds that mathematical statements have lasting 'objective' meanings.<ref>{{cite web | website=University of Fribourg | url=http://perso.unifr.ch/rafael.nunez/reviews.html | title=Where Mathematics Comes From |  url-status=dead | archive-url=https://web.archive.org/web/20060716051333/http://perso.unifr.ch/rafael.nunez/reviews.html | archive-date=July 16, 2006}}</ref> For example, [[Fermat's Last Theorem]] means exactly what it meant when [[Fermat]] initially proposed it in 1664. Other reviewers have pointed out that multiple conceptual strategies can be employed in connection with the same mathematically defined term, often by the same person (a point that is compatible with the view that we routinely understand the 'same' concept with different metaphors). The [[metaphor]] and the conceptual strategy are not the same as the formal [[definition]] which mathematicians employ. However, ''WMCF'' points out that formal definitions are built using words and symbols that have meaning only in terms of human experience.

Critiques of ''WMCF'' include the humorous:
{{bq|It's difficult for me to conceive of a metaphor for a real number raised to a complex power, but if there is one, I'd sure like to see it. — Joseph Auslander<ref name=Sutcliffe>[http://www.sutcliffe.com/blog/?m=200709 What is the Nature of Mathematics?], Michael Sutcliffe, referenced February 1, 2011</ref>}}
and the physically informed:

{{bq|But their analysis leaves at least a couple of questions insufficiently answered. For one thing, the authors ignore the fact that brains not only observe nature, but also are part of nature. Perhaps the math that brains invent takes the form it does because math had a hand in forming the brains in the first place (through the operation of natural laws in constraining the evolution of life). Furthermore, it's one thing to fit equations to aspects of reality that are already known. It's something else for that math to tell of phenomena never previously suspected. When Paul Dirac's equations describing electrons produced more than one solution, he surmised that nature must possess other particles, now known as antimatter. But scientists did not discover such particles until after Dirac's math told him they must exist. If math is a human invention, nature seems to know what was going to be invented.<ref name=Sutcliffe/>}}

Lakoff and Núñez tend to dismiss the negative opinions mathematicians have expressed about ''WMCF'', because their critics do not appreciate the insights of cognitive science. Lakoff and Núñez maintain that their argument can only be understood using the discoveries of recent decades about the way human brains process language and meaning. They argue that any arguments or criticisms that are not grounded in this understanding cannot address the content of the book.<ref>{{Cite web |last1=Lakoff |first1=George |author-link1=George Lakoff |last2=Núñez |first2=Rafael E. |author-link2=Rafael E. Núñez |title=Where Mathematics Comes From – Warning |url=http://www.unifr.ch/perso/nunezr/warning.html |archive-url=https://web.archive.org/web/20020613231421/http://www.unifr.ch/perso/nunezr/warning.html |archive-date=2002-06-13}}</ref>

It has been pointed out that it is not at all clear that ''WMCF'' establishes that the claim "intelligent alien life would have mathematical ability" is a myth. To do this, it would be required to show that intelligence and mathematical ability are separable, and this has not been done. On Earth, intelligence and mathematical ability seem to go hand in hand in all life-forms, as pointed out by [[Keith Devlin]] among others.<ref>{{citation|first=Keith|last=Devlin|title=The Math Instinct / Why You're a Mathematical Genius (Along with Lobsters, Birds, Cats and Dogs)|year=2005|publisher=Thunder's Mouth Press|isbn=1-56025-839-X|url-access=registration|url=https://archive.org/details/isbn_9781560258391}}</ref> The authors of ''WMCF'' have not explained how this situation would (or even could) be different anywhere else.

Lakoff and Núñez also appear not to appreciate the extent to which [[intuitionist]]s and [[Constructivism (mathematics)|constructivist]]s have presaged their attack on the Romance of (Platonic) Mathematics. [[Luitzen Egbertus Jan Brouwer|Brouwer]], the founder of the [[intuitionist]]/[[Constructivism (mathematics)|constructivist]] point of view, in his dissertation ''On the Foundation of Mathematics'', argued that mathematics was a mental construction, a free creation of the mind and totally independent of logic and language. He goes on to criticize the formalists for building verbal structures that are studied without intuitive interpretation. [[Symbolic language (mathematics)|Symbolic language]] should not be confused with mathematics; it reflects, but does not contain, mathematical reality.<ref>{{citation|first=David M.|last=Burton|title=The History of Mathematics / An Introduction|edition=7th|publisher=McGraw-Hill|page=712|year=2011|isbn=978-0-07-338315-6}}</ref>

Educators have taken some interest in what ''WMCF'' suggests about how mathematics is learned, and why students find some elementary concepts more difficult than others.

However, even from an educational perspective, WMCF is still problematic. From the conceptual metaphor theory's point of view, metaphors reside in a different realm, the abstract, from that of 'real world', the concrete. In other words, despite their claim of mathematics being human,  established mathematical knowledge — which is what we learn in school — is assumed to be and treated as abstract, completely detached from its physical origin. It cannot account for the way learners could access to such knowledge.<ref>{{Cite book|title=Mathematics and the body : Material entanglements in the classroom|last1=de Freitas|first1=Elizabeth|last2=Sinclair|first2=Natalie|publisher=Cambridge University Press|year=2014|location=NY, USA}}</ref>

WMCF is also criticized for its monist approach. First, it ignores the fact that the sensori-motor experience upon which our linguistic structure — thus, mathematics — is assumed to be based may vary across cultures and situations.<ref name=":0">{{Cite journal|last1=Schiralli|first1=Martin|last2=Sinclair|first2=Natalie|date=2003|title=A constructive response to 'Where mathematics comes from'|journal=Educational Studies in Mathematics|volume=52|pages=79–91|doi=10.1023/A:1023673520853 |s2cid=12546421 }}</ref> Second, the mathematics WMCF is concerned with is "almost entirely... standard utterances in textbooks and curricula",<ref name=":0" /> which is the most-well established body of knowledge. It is negligent of the dynamic and diverse nature of the history of mathematics.

WMCF's logo-centric approach is another target for critics. While it is predominantly interested in the association between language and mathematics, it does not account for how non-linguistic factors contribute to the emergence of mathematical ideas (e.g. See Radford, 2009;<ref>{{Cite journal|last=Radford|first=Luis|date=2009|title=Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings|journal=Educational Studies in Mathematics|volume=70|issue=2 |pages=111–126|doi=10.1007/s10649-008-9127-3 |s2cid=73624789 }}</ref> Rotman, 2008<ref>{{Cite book|title=Becoming beside ourselves : the alphabet, ghosts, and distributed human being|last=Rotman|first=Brian|publisher=Duke University Press|year=2008|location=Durham}}</ref>).