La prima sensazione che ho avuto leggendo questo libro è il famoso detto "quando uno ha in mano un martello, vede ovunque chiodi". Lakoff è un cognitivista: per lui dunque la matematica non può che essere il risultato delle connessioni neurali umane, se parliamo a basso livello, o delle analogie che noi umani facciamo se parliamo ad alto livello (Gli autori preferiscono il termine "metafore", ma il concetto di base è lo stesso). Il problema che vedo io è che è indubbio che noi ci facciamo delle idee mentali sui concetti matematici, ma questo non significa che i concetti siano il risultato di queste metafore. Pensiamo per esempio ai numeri immaginari e complessi: sono nati seguendo un certo tipo di formalismo ("facciamo finta che esistano e si comportino come i numeri usuali"), ma poi la metafora è mutata ("l'unità immaginaria corrisponde a una rotazione antioraria di 90 gradi nel piano cartesiano") senza che le proprietà cambiassero: banalmente, ora vediamo gli stessi oggetti in un altro modo più semplice. Gli autori partono dai (minimi) risultati cognitivi ottenuti a proposito della matematica, o meglio sui concetti come la subitizzazione e le somme di piccoli numeri; da lì costruiscono una cattedrale di filosofia della matematica, affermando che tutte le correnti attualmente esistenti, dal platonismo al formalismo al costruttivismo alla matematica sociale, non colgono la vera essenza della matematica. Occhei, atteggiamenti di questo tipo sono la norma in filosofia, quindi non c'è da stupirsi nel trovarli; ma da qui ad affermare di avere trovato la Verità ce ne corre. Un platonico standard quale io sono pensa che i concetti matematici esistano "da qualche parte", ma che non sono mappati perfettamente sul nostro mondo empirico; quindi tutte le pagine che gli autori usano per "dimostrare" che il continuo non è continuo e che per esempio gli infinitesimi hanno pieno diritto di esistenza sono per me puri esercizi intellettuali. Detto questo, è comunque importante leggere un punto di vista diverso dall'usuale e bene argomentato, anche per chiarirsi meglio le idee. ( )

George Lakoff is a national treasure - who through is career and many collaborations has provided that evidence of the inevitability and importance of metaphor. Metaphor structures how we reason - via the entailing logic of the frame, metaphor and narrative. For anyone interested in seeing the invisible metaphors we embody in our systems of logic and mathematics - this is a MUST READ. ( )

Interesting follow-up to Lakoff's Metaphors We Live By.
The thesis here is that all language is metaphorical expression, which is based on conceptual metaphors determined by the brain as well as by society.
Mathematical metaphors are explained within this context, starting with things like the number-line and the Cartesian plane and culminating in the example of Euler's equation where "e to the i pi plus one equals zero". Quite a ride and well worth skimming through the tables and the chapter summaries in order to get a feel of how concept work whithin our brains. ( )

This book introduces a new sub-discipline of cognitive science called “mathematical idea analysis”. In brief this sub-discipline seeks to provide a more intuitive understanding of many abstract mathematical ideas by tying those ideas to the often unconscious understanding of our physical environment (embodiment). From the preface:

“Human ideas are, to a large extent, grounded in sensory-motor experience. Abstract human ideas make use of precisely formulatable cognitive mechanisms such as conceptual metaphors that import modes of reasoning from sensory-motor experience. [ … ] The central question we ask is this: How can cognitive science bring systematic scientific rigor to the realm of human mathematical ideas, which lies outside the rigor of mathematics itself? Our job is to help make precise what mathematics itself cannot—the nature of mathematical ideas.” Pg. XII

The book is creative, ambitious, and well organized. For the most part the writing is good, although I thought it became repetitive at times where the authors would start a new section by repeating several ideas from the last section. Some people may like this style - it’s very structured and “scientific” - but it also gets tiring, especially in a 450 page book.

I think overall the book is a mixed bag. The introductory chapters and the summary chapters (on the theory and philosophy of embodied mathematics) were pretty good and I enjoyed them. But the majority of the books middle chapters are focused on a very detailed construction of all the metaphors needed for mathematical idea analysis. There are countless tables making detailed metaphorical mappings from “source domain” to “target domain”. These often seem trivial or obvious and I kept thinking to myself ”Well, someone’s got to do this but I’m not really interested in reading about it!” (I imagine the same thing has been said of Russell and Whitehead’s “Principia Mathematica”, it was an amazing achievement but not great reading). My feeling is that only specialists in cognitive science or mathematical pedagogy would find these chapters useful.

Overall I wouldn’t recommend this book for a general reader wanting to learn more about mathematics or cognitive science since I think there are better books on these subjects. For instance, if you are interested in cognitive science and the idea of embodiment I highly recommend Lakoff’s earlier book “Women, Fire, and Dangerous Things”. If you already own "Where Mathematics Comes From" then reading chapters 1, 2, 15 and 16 should give you a good feel for the book and whether you want to continue or not. ( )

I've never ready anything about cognitive science and as this book is a look at Mathematics from the Cognitive Scientist point of view it was difficult to start. By the end of the book I was pretty enthralled. Any one that has taken some higher level math courses (analysis, abstract alg) should read this book and really think about what they've learned. Anyone planning on teaching higher level math should read this and think about how they teach.