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Metaphors and errors play roles in various types of mathematical and physical reasoning.

Consider the following statement by the well known French mathematician, Alain Connes (Triangle of Thoughts,  2001, p. 26) 

“A mathematical object has an existence that is every bit as solid as external reality.”

In essence it is the view that Plato  and many Greeks affirmed about 2400 years ago: that there is a place where mathematical thoughts exist independent of the human mind. Many mathematicians from Plato to Godel to Connes have posited such a view , and still do. They believe in a  Platonic heaven where our  mathematical thoughts have an existent reality.
I call this ‘The Platonic Fallacy’.

An instrumental metaphor in forming this metaphysical posture is Plato’s allegory of the cave where he conceived  of reality as viewed by us through the shadows it forms on the wall of a cave --the ‘ding an sich’ is therefore only known indirectly.  I believe  such  Platonists are captured by  metaphoric equivalents of the cave allegory that links the real world and our perceptions of it with the ‘world of ideas’, as though, when it is said, ’what do we mean by the number 3’ it is as though the number 3 exists--when in truth the number 3 is part of a metaphoric network embodied in our mind/brain structures.

By the word metaphor I mean an extension of the traditional literary trope to include that which stands for something else, is isomorphic to it, is a mapping of it, a correspondence, or a gedanken experiment. Metaphors permeate our mind/brain complex and are part and parcel of all our thought processes.  Metaphors and their networks reach into deeper layers of our cognitive structures than symbols.

A root paradigm of a metaphor in a developmental process is seen when we point at an object to induce a child about two years of age to look at it. The child generally looks at our finger and only after some time turns to look at the object-- the child is forming the metaphor, "don’t look at the finger but look to where the finger is pointing".

The Platonic Fallacy is an error in which the ideational object or the mathematical idea is thought to exist in an imagined world, instead of being seen as constructed by and embodied in the mind/brain. The fallacy is made more manifest when new mathematical ideas or theorems are characterized as being ‘discovered’ ( in another world) rather than invented ( in our mind/brain)

I would now like to flesh out the notion of a metaphor/error dialectic in our knowing processes and deepen the sense that these knowing processes and what we know are embodied in our mind/brain,  in the following examples:

A root of the aphorism attributed to the ancient Greeks, that nature abhors a vacuum, may be found in Aristotle's physics, where he analyzes the motion of a falling body. Aristotle said, in effect, that an object falls slowly in a highly viscous medium, more quickly in thinner oil, and even more quickly in air. He concluded that the speed of an object is inversely proportional to the viscosity of the medium in which it falls. Ergo an object would fall with infinite speed in a vacuum (which has zero viscosity), which is impossible; therefore there can be no vacuum. (The famous dictum was "Nature abhors a vacuum.")

This kind of error occurs in scientific literature up to the 17th century and is found in Kepler's writings before his elliptical orbit theory appears. Kepler, while investigating the motion of Mars with respect to the Sun, noted that its speed at its furthest point from the Sun is slower than its speed at its closest point. He concluded erroneously that the speed of Mars was inversely proportional to its distance from the sun. Students in today's schools (including colleges), when presented with Aristotle's and Kepler's analyses, without being forewarned that they are both in error, generally agree with their analyses and conclusions. In other words, both historically and psychologically, assumptions about numerical connections among things tend to be linear or inversely linear. This suggests a psychological preference for certain modes of cognition that span the centuries. We have exhibited the metaphor of linearization (or inverse linearization).  Furthermore, among sophisticated scientists this kind of error does not occur after the 17th century.

A further illustration of this metaphor/error dialectic is the following:

In 1905, Einstein’s miracle year,  Einstein laid down foundations for physics for the indefinite future and also presented us  a mystery which may remain. In a dazzling display of insight he published a number of papers in the Annalen der Physik  and  an afterthought  to these which is touched with mystery. Part of the mystery is what must have been in his mind as he prodigiously juggled a host of metaphors about the nature of things, including many gedanken experiments, while working full time in the Swiss patent office in Berne. But the mystery we reach for involves the last paper of the annus mirabilis, and what forms the paradigm and a clarification of Einstein's  famous dictum, "imagination is more important than knowledge," which he experienced profoundly in this year.

With  deep inward forces and fruited metaphors of  that year, 1905, and with his writings on "Brownian Motion," Einstein solidified the sense of the reality of the atom.  With "the photoelectric effect" he foreshadowed the quantum theory and the move away from continuity on the atomic level towards a discrete universe.  With the "Special Theory of Relativity" he  paradoxically fostered a continuous world and the sense of a new post-Newtonian world geometery--contra Euclid.

The last paper, stemming from the Special Theory, was what would appear to be an afterthought, coming some months later.  How he juggled metaphors  that must have burned in his mind, which spoke of the equivalence between matter and energy.  Thoughts connecting the two had been touched on in the previous ten years by J.J. Thomson and W. Kaufmann. His mass-energy paper posited a quantitative equivalence, E=MCsquared.

In an imaginative burst, using a gedanken experiment, rather than with a demonstrative development, Einstein, with an approximation of his relativistic results, used the binomial theorem to indicate that the energy radiated  (call it E) by an object moving with velocity ,v,  might be thought of as equivalent to a quantity called 1/2 (glunk)Vsquared, which I believe he thought of metaphorically  as kinetic energy.  That glunk which was E/Csquared  (where C=speed of light),   was where an object's mass (call it M) usually is when we speak of kinetic energy.  His astounding imaginative rendition was then read as E=MCsquared. Where he thought of M as the mass equivalent of the radiational energy of the moving object.   Einstein then said, and so do many others, that he had demonstrated  the equivalence of mass and energy, but he hadn't.  He effectively assumed what he wanted to demonstrate. (That there was a mass equivalent of the radiational energy).

We know about the mystery: Why did Einstein not realize that he had made  an imaginative leap which he then went on to solidify into a principle of physics over the next two  years?   The answer may be that Einstein was so captured by the metaphors of the equivalence and that what looked like kinetic energy was kinetic energy, that his mind, highly wrought with the amazing variety of metaphors he had been producing, could not see in 1905 what others, like Max Jammer, in The Nature of Matter,  subsequently realized: Einstein did not prove the equivalence in 1905!

A last example which shows both the power and the beauty of the metaphoric basis of mathematical thought will be in the fruitfulness of the number /geometry dialectic. Incidentally, this dialectic is the basis of analytic geometry. Number and geometry systems can function as metaphors for each other. This example connects various geometries and various number systems:  In many geometries, Euclidean, projective etc., one or both of Pappus’ and Desargues’ theorems are true . In addition, one can transform the geometry into a number system in which the properties of addition and multiplication are defined geometrically. We will not do the complicated analysis, but it can thereby be shown that the geometry and the connected number system are isomorphic--- that they are metaphors for each other!

It can also be shown (we will not do this either) that Pappus’ theorem in the geometry is equivalent to the number structure being commutative under multiplication ---  and  that Desargues’ theorem is equivalent to the numbers being commutative under addition.

We will now show that, generally speaking, (particularly if a number system has a multiplicative identity) commutativity under multiplication implies commutativity under addition. See proof A below. (The converse is not true as one can see with matrices-- they are not commutative under multiplication, but are so under addition). Using this metaphoric connection between the numbers and the geometry we then have the beautiful and powerful theorem that Pappus’ theorem implies Desargues’ theorem. This would be a formidable proof had we remained in the geometric system and had not shifted metaphorically to the number system!

Proof A (multiplicative commutativity implies additive commutativity)
1.  (a+b)(c+d)= (c+d)(a+b)
2. (a+b)c +(a+b)d =(c+d)a +(c+d)b
3. ac+bc +ad+bd =ca+da+cb+db
4. therefore, bc+ad=ad+bc
5. allowing c and d to be 1 we then have
6.b+a=a+b --(additive commutativity).   QED