This is a more serious philosophical discourse than other theses, or mini-theses, I’ve posted so far. It’s an argument I’ve had with a number of philosophers, and non-philosophers. It's a question that most philosophers, indeed most people, seem to have an opinion on.
The short answer is that it’s a mixture of both invention and discovery. Mathematics requires creativity to achieve breakthrough discoveries as does any field of science. But I’m short-circuiting the argument. A good starting point is to reference a book I’ve read, Where Mathematics Comes From, by George Lakoff and Rafael E. Nunez. This is an excellent book on mathematics, covering all the basics and a number of esoteric topics like calculus, transfinite numbers and Euler’s famous equation: eiΠ + 1 = 0. This equation brings together such diverse fields as trigonometry, logarithms, calculus, complex algebra and power series into one simple relationship. The physicist, Richard Feynman, who discovered the equation a month before his 15th birthday, called it ‘the most remarkable formula in math’. Lakoff and Nunez provide a very accessible derivation of this equation as the crowning piece of exposition in their book. I must say at the outset that I have neither the expertise nor the ability to write a book like this. It is a very good book on mathematics. All my arguments and contentions deal with its philosophical content.
Lakoff and Nunez eschew any notion that mathematics is ‘discovered’, which is not an uncommon position. They argue, reasonably enough, that mathematics can only come from an ‘embodied mind’, therefore any suggestion that mathematics ‘already exists a priori’ is a conundrum that defies rational explanation. They argue that the only mathematics we know of comes from the human mind, therefore the onus of proof for any alternative view rests with the proponent of that view. In other words, the default point of view has to be that mathematics only exists as a product of the human mind. There is no evidence to support any other point of view.
Just to address that last point: all scientific discoveries are products of the human mind, nevertheless they exist independently of the human mind as well. The specific problem with giving mathematics the same status is that it doesn’t exist materially independently of the human mind. I will come back to this point later.
But my main problem with Lakoff’s and Nunez’s book is the assertion that all mathematics can be explained by ‘conceptual metaphor’. I’ve since learned that this particular philosophical premise is a brainchild of George Lakoff’s, who has written numerous books explaining the significance of metaphor in human endeavour, including philosophy, science, and, of course, mathematics. George Lakoff is Professor of Linguistics at Berkeley University, and I’ve since had correspondence with him. I’ve come to the conclusion that we agree to disagree, though he never responded to my last correspondence.
Many of my criticisms of Professor Lakoff’s philosophy addressed in this blog (though not all) have been made to him directly. In his book, Philosophy in the Flesh, which he co-wrote with Mark Johnson (not Nunez), Lakoff seems to find fault with every philosopher he’s acquainted with, both living and dead. He does this by employing his own 'philosophy of metaphor' (my terminology, not his) to give the reader his interpretation of their ideas. Much of this posting deals with Lakoff's use of the word metaphor. Its relevance to mathematical epistemology is explained in the next paragraph.
Basically, a conceptual metaphor ‘maps’ from a ‘source domain’ to a ‘target domain’ to use Lakoff’s own nomenclature. In the case of mathematics, the source domain is the grouping of objects, and activities that involve removal or combining elements of groups or complete groups in various ways. The target domain are the concepts inside our heads, which we call numbers, and how we manipulate them to represent events in the real world. Target domains can also be graphical representations like number lines and geometrical figures. This is not a verbatim representation of Lakoff’s and Nunez’s ideas, but my interpretation to ensure brevity of exposition without losing the gist of their philosophical premise. I have no problem with this aspect of their argument. I agree that mathematics is one of the most efficacious mediums we have for bridging the external world with our internal world. I have previously explained that the experiential concept of the external and internal world seems to be the starting point for many of my philosophical discourses. Where I disagree with Lakoff and Nunez is their assertion that this ‘bridge’ is strictly metaphorical.
According to The Oxford Companion to the Mind, metaphor is determined by context. This definition of a metaphor assumes that a word, phrase or term that is used in a metaphorical context must also have a literal context. In the case of Lakoff’s conceptual metaphors, that comprise all of mathematics, the metaphorical and literal contexts appear to be the same. I asked Professor Lakoff: ‘In what context is 2+2=4 metaphorical and in what context is it literal? If I say I want 3 of those, am I talking metaphorically and literally at the same time?’ The impression I got from his book is that mathematics has no literal context, only a metaphorical context. In other words, with ‘conceptual metaphors’, he has created a whole new field of metaphors that are permanently metaphorical. I can see no other interpretation and Lakoff has failed to enlighten me when I challenged him specifically on this. Assuming my interpretation is correct, this begs the question: they are metaphors for what? The obvious answer, going back to the original ideas set out in the ‘source domain’ and the ‘target domain’, is that they are metaphors for reality.
In Philosophy in the Flesh, Lakoff continually talks about metaphor as if it’s the progenitor of all ideas and concepts. He analyses a philosophical idea by reducing it to metaphor then presents it as if the metaphor came first. I will discuss an example that’s relevant to the topic: time and space. Lakoff rightly expounds on how we often use terms associated with distance to talk about time – it’s like we visualise time as distance. In relativity theory, this visualisation is real, due to a peculiar property of light. In ancient cultures and some indigenous cultures, however, the reverse is true: they refer to distance in terms of time. When Eratosthenes calculated the earth’s circumference around 230BC, he measured the distance he traveled from the well in Syene (Aswan) to Alexandria by the number of days he traveled by camel. If this was a metaphor and not literal then his whole enterprise would have failed. As it was, his calculation of the earth’s circumference was out by 15% according to modern measurements (ref: Encyclopaedia Britannica).
Everyone knows that there is a mathematical relationship between distance, time and speed, which is literal and not metaphorical. Now all physicists know that this relationship breaks down at sub-atomic speeds and astronomical distances due to relativity, so how can we say it’s true or literal or real? To add a further spoke in the works, when we have quantum tunneling the relationship ceases to exist altogether. But these anomalies are not resolved by saying that they are all metaphors and not real. They are resolved by finding the correct mathematical relationships that nature follows in these circumstances.
Physicists like Roger Penrose and Paul Davies have written extensively on the remarkable concordance we find between mathematics and the physical world. Lakoff claims that this concordance is purely metaphorical, and by his definition of metaphor (source domain: events in real world; target domain: concepts in our heads) I would agree. Using Lakoff’s own logic, mathematics is a metaphorical representation of the real world, but in this use of the term metaphor there is no distinction between metaphorical language and literal language – metaphor is a direct translation. Lakoff often uses the term metaphor where I would use the word definition. When he defines a concept in terms of other known concepts he calls it a conceptual metaphor or a conceptual blend. Conceptual blend is bringing 2 or more concepts together to form a new concept. Conceptual blend makes sense, but conceptual metaphor doesn’t if there are no distinct literal and metaphorical contexts in which to make it a metaphor. I’ve also argued that where there is a causal relationship between 2 concepts, one is not necessarily, by default, a metaphor for the other. An example of this is periodicity being a direct consequence of rotation; day and night resulting from the earth’s rotation is the best known example. In Where Mathematics Comes From, Lakoff implies that this relationship is metaphorical.
Personally, I call Lakoff’s conceptual metaphors literal metaphors, because if they were literal then my entire argument on this issue would evaporate, which, of course, would be preferable for both of us.
Lakoff also maintains that all theories (in physics at least) are metaphorical, which is not an issue I will pursue here. I did point out to him, however, that some of his metaphorical interpretations (of Einstein’s theories in particular) were incorrect or misleading. I referenced Roger Penrose, who is more knowledgeable on this subject than either of us.
Lakoff argues that physics is effectively mathematical modeling that happens to get very close to what we observe, and there are many who would agree with this interpretation. (Renowned physicist, Stephen Hawking, subscribes to this view.) But many physicists would say that the mathematical concordance we find in nature goes beyond modeling because there are too many cases where the mathematics provides an insight into nature that we didn’t expect to find (for example: Maxwell’s equations giving us the constant speed of light in a vacuum or Dirac’s equation giving us anti-matter). Irrespective of this argument, nature follows mathematical relationships at all observable levels of scale. Lakoff, by the way, argues that there are no ‘laws of nature’, which is another argument, though not altogether irrelevant, that I won’t pursue here.
This has been a lengthy detour, but it brings me back to the point I made about the status of mathematics existing independently of the human mind. Most people struggle with this notion – it’s like believing in God. It evokes the idea of an abstract realm independent of human abstract thought. People call this the Platonic realm after Plato’s fabled realm of perfect ‘forms’. The real world, in which we live, being a shadow of this perfect transcendental world. Roger Penrose calls himself a Platonist (refer The Emperor’s New Mind), mathematically speaking, because he believes the mathematics we discover already exists ‘out there’. Paul Davies eschews the idea of Platonism (refer The Goldilocks Enigma) but in The Mind of God, he devotes a whole chapter to what he calls ‘the mathematical secret’: the way the physical world is ‘shadowed’ by mathematics.
I call myself a Pythagorean because Pythagoras was the first (in Western philosophy at least) who seemed to appreciate that mathematics is an inherent aspect of the natural world, like a latent code waiting for someone like us to decipher it. As our knowledge of physics progresses, this realisation only seems to become more necessary to our comprehension of the universe. I can’t help but feel that Pythagoras had no idea how deep his insight really was. Lakoff calls Pythagoras’s philosophical insight a ‘folk theory’, but it’s a folk theory that launched Western science as we know it, so I would call it a paradigm, and one that has had unparalleled success over a 2,500 year history.
Philosophers, since the time of Russell and Wittgenstein, have mostly argued that mathematics is a sub-branch of logic, but Godel, Turing and Church, have all demonstrated, in various ways, that one cannot create all mathematics from a set of known axioms (Godel’s famous incompleteness theorem). This is one point that Lakoff and I appear to agree upon.
What many people fail to understand, or take into account, is that mathematics is not so much about numbers but the relationship between numbers – just look at how much mathematics is written without numbers. Robyn Arionrhod, who teaches mathematics at Monash University, Melbourne, made a similar point in her book, Einstein’s Heroes (effectively, a well written exposition on Maxwell’s equations). If one looks at mathematics from this perspective, one can see that the relationships cannot be invented – they are discovered. Mathematics is essentially a problem-solving endeavour, and this begs the question: if one is looking for a solution to a puzzle, does the solution already exist before someone finds it, or only after it’s found? Think: Fermat's Last Theorem; solved by Andrew Wiles 357 years after it was proposed. Think: Poincare's Conjecture; proposed 1904 and solved in 2002 by Grigory Perelman (read Donal O'Shea's account). Think: The Reimann Hypothesis; proposed 1859, still unsolved. To answer that question, I suggest, is to answer the philosophical conundrum: is mathematics invented or discovered?
Footnote: I sent a copy of this to Professor Lakoff as soon as I posted it, offering him the right to reply.
For an alternative point of view, read Lakoff’s and Nunez’s book, Where Mathematics Comes From. For a physicist’s perspective, read Penrose’s The Emperor’s New Mind or Davies’ The Mind of God. Arionrhod’s Einstein’s Heroes is a good read that indirectly supports the physicist’s perspective.
See also my later posting, Jan. 08: Is mathematics evidence of a transcendental realm? Amongst other things, I discuss Gregory Chaitin's book, Thinking about Godel and Turing.