Mathematics is an introspective science, as opposed to experimental sciences(Physics, Chemistry etc). Unlike experimental sciences, progress in mathematics doesn’t require getting data from deep sky, probing into an atom or colliding elementary particles at high speeds. Numbers and other concepts are in our head and all open problems are in our head. This suggests a possibility, that even if we locked ourselves in a cave, cutting off all communications with the nature, we may still be able to continue our progress in mathematics. The subject of this blog post is to examine if this is really possible.
The converse of this question has a clear answer: Halting mathematical progress would halt progress in all experimental sciences. Mathematics develops ways of thinking, that are employed in understanding the nature. It appears that mathematics is on its own; it doesn’t depend on any of the other sciences. It can continue progressing without any other sciences. This view is expressed in this xkcd comic. But I am going to contradict this view (and I am going to contradict other aspects of that cartoon in my next blog post :D).
Although it is not apparent, progress in mathematics requires us to experimentally probe into nature. The ideas involved in mathematical research are seeded by our experimental probes. Exploration of nature is a key not only to develop physical sciences, but also to develop mathematics, although it is a purely logical subject. A problem is deemed solved, only after a logically consistent solution is found. However, the question “what is an interesting problem to solve?” doesn’t quite have a logical answer. G. H. Hardy has described this question to be akin to asking where do poets, writers and other artists get ideas for their work. Conceiving new problems is a work of imagination. And imagination is always seeded by reality.
The classic example for an interesting problem is the one that led to Fermat’s last theorem: ‘are there integers a, b and c such that an+bn = cn for some integer n?’. Surely, the complete answer to this problem was profoundly useful, only because large number of people worked on it and it developed a great deal of understanding of numbers. However, the problem was presumably seeded by the Pythagoras theorem. We have integers, like (3,4,5) such that 32+42=52, so a curious question like ‘what if we change the exponent to a number other than 2?’, would have been the origin of Fermat’s last theorem.
Another question in mathematical research that doesn’t have a logical answer is “what set of axioms should we choose?”. Axioms, like problems, are chosen by taking a cue from previous mathematical theories. The resulting structural similarity between different mathematical theories has been capitalized in a theory called category theory.
Where do these ‘previous mathematical theories’ get their axioms and problems from? There must be a starting point, a seed for every mathematical idea. These seeds come from outside- from our interaction with nature. Cutting off interaction with nature will cut off the supply of new seeds. But that doesn’t entirely stop mathematical progress; instead, ideas for new mathematical theories will be entirely dependent on the old mathematical theories. Over a timescale of several hundred years, this is a significant setback to mathematical progress. Seeding of mathematics by interaction with nature is a slow process. In fact, we are still benefiting from the seeds of Pythagoras theorem.
The seeds of Pythagoras theorem
‘Geometry’ stands for measurement of the earth. ‘Earth’ here doesn’t mean the planet earth or the globe; it means land; real estate. Geometrical ideas were developed as a result of extensive measurement of land, during early human settlements. The most influential of these was the Pythagoras theorem. Let me go through its development, in its three chronological stages: the content, the statement and the proof, to identify its seeds.
Given two sides of a right triangle, knowing how to calculate the third side is the essential content of Pythagoras theorem. This may be done using a formula, or using tabulated data or using similarity of triangles. All these methods carry the basic wisdom– “the third side of a right triangle is not an independent variable”. Any civilization that built large planned settlements knew the content of Pythagoras theorem.
Explicit statements probably came several thousand years after the content. Early statements of Pythagoras theorem were in terms of areas. There are records of statements in Babylonian scriptures(2000 BC), in the Vedas(Shulva Sutras, 800 BC) and Chinese scriptures. An explicit statement could’t have brought any change in the applicability. Perhaps, the room/house in which the statements were written was constructed using the content of Pythagoras theorem :D. However, it brought big changes in theory. Geometrical shapes were understood by cutting them into triangles. Triangles were now understood by cutting them in to right triangles. Right triangles took precedence over other triangles, leading to a new branch- Trigonometry.
A partial proof was recorded in 800 BC(Shulva Sutras) and a complete proof in 500 BC(Pythagoras). Presumably, there were unrecorded proofs prior to this. There are some theories that Pythagoreans might have been communicating with Chinese schools of mathematics. The proof was seeded by two pieces of intuition, which were developed when planning settlements. One is that land can be measured in areas, which can be added and subtracted by joining and cutting pieces of land. The second is scaling; a big piece of land can be scaled down and represented on paper or a flat stone. All proofs of the Pythagoras theorem are based on areas of triangles or scaling of small triangles to big ones(A book, The Pythagorean Proposition lists 370 proofs). Scaling is in fact a logical implication of the properties of areas. But it is likely that it was developed independently.
The proof, of course, had no practical implications. Proofs generally store methods of thinking. Even today we feel its impact on our thinking. The proof raised the status of the mere formula, ‘a2 + b2=c2′ to a theorem, resulting in the discovery of irrational numbers. Furthermore, it redefined the whole of geometry in terms of a single quantity- the distance between two points. The more advanced forms of geometry- Riemannian geometry and even Differential geometry contain the germs of Pythagorean distance.
The Pythagoras theorem and all of its intellectual impact on Mathematics are seeded by man’s physical exploration in to measuring land. A writer, within his lifetime travels extensively to gain experiences of reality that can seed his imagination. Mathematics is also seeded by explorations of real world, but this seeding has a longer timescale, much longer than a mathematician’s lifetime.