Is mathematics invented or discovered? Do mathematical objects pre-exist in some transcendental plane, are they abstractions of our sensory experiences, or are they just fictional objects invented by our minds? Would an alien species specify mathematical abstractions in a different way? These are the questions which are explored under the “Philosophy of mathematics”. Several competing theories such as Logicism, Intuitionism, Formalism and Platonism have been proposed to explain the nature of mathematics. Here, I shall present some insights by Sri Aurobindo and the Mother which are connected to this topic.
Mathematicians who get breakthroughs in flashes of intuition solemnly aver that mathematical objects must pre-exist in some transcendental plane. Srinivasa Ramanujan claimed that the Goddess Namagiri would visit him in dreams and give him those astounding equations that continue to boggle the minds of mathematicians. Paul Dirac said that it was his keen sense of beauty that enabled him to divine the Dirac equation for the electron which others had searched in vain. The renowned Hungarian mathematician Paul Erdos used to say that there was a “Book” in which God kept the most elegant proofs of all theorems. He wasn’t religious per se but this was his way of expressing the sheer delight that he and other mathematicians experience whenever they come across an ingenious mathematical solution. Jacques Hadamard stated that the moment of breakthrough felt similar to the moment when we suddenly recognize someone’s face. The French mathematician Andre Weil expressed it poetically when he wrote:
«Rien n’est plus fécond, tous les mathématiciens le savent,
que ces obscures analogies,
ces troubles reflets d’une théorie à une autre,
ces furtives caresses,
ces brouilleries inexplicables:
rien aussi ne donne plus de plaisir au chercheur.
Un jour vient où l’illusion se dissipe;
le pressentiment se change en certitude
les théories jumelles révèlent leur source commune avant de disparaître Comme l’enseigne la Gita on atteint a la connaissance et a l’difference en meme temps
La metaphysique est devenue mathematique, prête à former la matière d’un traité dont la beauté froide ne saurait plus nous émouvoir »
“Nothing is more fruitful—all mathematicians know it—
than those obscure analogies,
those disturbing reflections of one theory on another;
those furtive caresses,
those inexplicable discords;
nothing also gives more pleasure to the researcher.
The day comes when this illusion dissolves:
the presentiment turns into certainty;
the yoked theories reveal their common source
As the Gita teaches, one achieves
knowledge and indifference at the same time
Metaphysics has turned into mathematics, ready to form the contents of a treatise whose cold beauty could no longer move us.
Is it really possible to characterize and qualify the beauty that mathematicians feel when they come across an elegant solution ?
In a recently published paper, researchers at the University College, London found that the experience of mathematical beauty activates the same region of the emotional brain — namely the medial orbito-frontal cortex — as the experience of beauty derived from art or music.
Professor Semir Zeki, lead author of the paper from the Wellcome Laboratory of Neurobiology at UCL, said: “To many of us mathematical formulae appear dry and inaccessible but to a mathematician an equation can embody the quintescence of beauty. The beauty of a formula may result from simplicity, symmetry, elegance or the expression of an immutable truth. For Plato, the abstract quality of mathematics expressed the ultimate pinnacle of beauty.”
“This makes it interesting to learn whether the experience of beauty derived from such as highly intellectual and abstract source as mathematics correlates with activity in the same part of the emotional brain as that derived from more sensory, perceptually based, sources.”
…”We have found that activity in the brain is strongly related to how intense people declare their experience of beauty to — even in this example where the source of beauty is extremely abstract. This answers a critical question in the study of aesthetics, namely whether aesthetic experiences can be quantified.”
“We have found that, as with the experience of visual or musical beauty, the activity in the brain is strongly related to how intense people declare their experience of beauty to be — even in this example where the source of beauty is extremely abstract. This answers a critical question in the study of aesthetics, one which has been debated since classical times, namely whether aesthetic experiences can be quantified.”
(Read the full report over at sciencedaily)
The general question of “Where does mathematics comes from?” is studied under the field called “Philosophy of Mathematics”. Philosophers have proposed many theories such as Platonism, Constructivism, Intuitionism, Logicism to explain the origin of mathematics. Platonists believe that mathematical objects reside in some transcendental plane and mathematicians are really just discovering them. Formalism, proposed by David Hilbert, claims that mathematical equations are just consequences of certain string manipulation rules. Logicism developed by Frege seeks to reduce all mathematics to pure logical rules. There are others who believe that mathematics is just an outcome of the cognitive activity of the human brain. (Aristotle was the first who proposed the germ of this theory).
An analogy might help us understand the differences between these theories. Consider an exotic object such as a Swiss Army knife. If you felt that the knife was so incredibly perfect that its design must pre-exist in some mythical world, then you are a Platonist. If you thought the knife was constructed using step-by-step rules, you are a formalist. If you felt that the knife evolved from the repeated trial-and-error experiments of many scientists, you are an empiricist.
We won’t enter deeply in this philosophical debate for now but we will present a few remarks by the Mother which are connected to this topic. Her views can be said to be aligned with Platonism and with the recent finding that mathematical beauty and artistic beauty awaken the same region of the brain.
Scientific and artistic imagination are complimentary
Question: How is it that in people occupied with scientific studies artistic imagination is lacking? Are these two things opposed to each other?
Mother: They do not belong to the same domain. It is exactly as though you had what is called “a torchlight”, a small beacon-light in your head at the place of observation. Scientists who want to do a certain work turn the beacon in a particular way, they always put it there and the beacon remains thus: they turn it towards matter, towards the details of matter. But people with imagination turn it upward, because up above there is everything, you know, all inspirations of artistic and literary things: this comes from another domain. It comes from a much more subtle domain, much less material. So these turn upward and want to receive the light from above. But it is the same instrument. The others turn it downwards, and it is just a lack of gymnastic skill. It is the same instrument. It is the same power of a luminous ray upon something. But as one has made it a habit of concentrating it in a certain direction, one is no longer supple, one loses the habit of doing things otherwise.
But you can at any time do both the things. When you are doing science, you turn it in one direction and when you do literature and art, you turn it in the other direction; but it is the same instrument: all depends on the orientation. If you have concentration, you can move this power of concentration from one place to another and in every way it will be effective. If you are occupied with science, you use it in a scientific way, and if you want to do art, you use it in an artistic way. But it is the same instrument and it is the same power of concentration. It is simply because people do not know this that they limit themselves. So the hinges get rusty, they do not turn any more. Otherwise, if one keeps the habit of turning them, they continue to turn. Moreover, even from the ordinary point of view, it is not rare to find a scientist having as his hobby some artistic occupation – and the reverse also. It is because they have found that the one was not harmful to the other and that it was the same faculty which could be utilised in both.
Essentially, from the general point of view, particularly from the intellectual viewpoint, the most important thing is the capacity of attention and concentration, it is that which one must work at and develop. From the point of view of action (physical action), it is the will: you must work and build up an unshakable will. From the intellectual point of view, you must work and build up a power of concentration which nothing can shake. And if you have both, concentration and will, you will be a genius and nothing will resist you.
(Collected Works of the Mother, vol 5, p 128-129)
Ideas comes from the Universal Mind
Mother: Ideas have a higher origin than the mind. There is a region of the mind, higher than the ordinary mind, in which there are ideas, typal ideas, really prototypes; and these ideas descend and are clothed in mental substance. So, in accordance with—how to put it?—the quality of the receiver, they either keep all their own qualities and original nature or become distorted, coloured, transformed in the individual consciousness. But the idea goes far beyond the mind; the idea has an origin much higher than the mind. So, the functioning is the same from both the universal and the individual point of view; the individual movement is only representative of the universal one. The scale is different, but the phenomenon is the same. Of course, these are no longer “thoughts” as we conceive thoughts; they are universal principles— but it’s the same thing—universal principles on which the universes are built.
In another conversation, she expounded on the different transcendental planes which are the source of creativity:
The first zone you encounter is the zone of painting, sculpture, architecture: everything that has a material form…They are not forms as we know them, but rather typal forms; you can see garden types, for instance, wonderfully colored and beautiful, or construction types.
Then comes the musical zone, and there you find the origin of the sounds that have inspired the various composers. Great waves of music, without sound. It seems a bit strange, but that’s how it is.
Beyond the musical zone lies thought: thoughts, organized thoughts for plays and books, abstractions for philosophies. That is the third zone. What you find there are thought formations that are expressed in each person’s brain in his own language. There are thought combinations for novels, plays, even philosophical systems. They are combinations of pure thought, not formulated in any language, but they are automatically expressed in each one’s brain according to his particular language. It is the domain of pure thought….
Higher up, there is a fourth zone, a zone of colored lights, plays of colored lights;… it is a zone of forces, a zone which appears as colored lights. No forms – colored lights representing forces. And one can combine these forces so that they work in the terrestrial atmosphere and bring about certain events. It’s a zone of action, independent of form, sound and thought; it is above all that. (Mother’s Agenda, 27 Oct 1962)
In other words, mathematical objects per se do not exist in some transcendental world but the generic patterns of thinking which influence our ideas do exist. These ideas descend into our mind through the four levels of Vedic Vak covered in an earlier article. Also check the articles on Roger Penrose versus Sri Aurobindo which explores this topic further.
- Roger Penrose and Sri Aurobindo on the Mind
- How to develop intuition
- Four Powers of Intuition
- Study of science as an aid in Yoga
- The role of intellectual development in the spiritual path
- How does the mind change with Yoga?
- All thoughts come from outside
- Distinguishing between stilling the mind and dynamizing meditation
- Syncretism in Sri Aurobindo’s thought – part 1
- Syncretism in Sri Aurobindo’s thought – part 2
- How does the brain absorb new ideas?
- Vedic Vak: illustration of Para Vak
- Brain imaging can reveal the movies in our mind
- The brain is not the mind as per Yoga psychology
- Perception of Time changes with the concentration of consciousness
- Difference between genius and mysticism