Rohit Dhankar
Recently I again came across a strongly made claim that making of mathematics is an experimental inductive science. As justification it is claimed that the practice of creating/making mathematics requires one to look at patterns, make a hypothesis, generalise, prove etc. The claim here is about “making of mathematics” or “actual practice of mathematics”. I think these phrases mean investigations and finding/creating mathematical knowledge. And then this claim is used to prepare a plan to teach mathematics, in which mathematics is seen as an experimental inductive science, not only in making but as it is and its pedagogy.
Are these claims true? I see many problems which might mislead mathematics teaching and therefore, worth paying attention for a teacher. Problems are mainly concerned with nature of mathematical knowledge and pedagogy of math.
The nature of mathematical knowledge:
As stated above, the claim is that creation of mathematics is an “experimental inductive science”. Yes, there is such a view that mathematics is a social activity in which the experience of concrete world is translated into abstract knowledge through induction and social (or community of mathematicians’) consensus. But the claims do not seem to be tenable on closer examination.
All knowledge formation is a social activity in the sense that it reflects social needs, it is used for social purposes, it’s creators are accorded social recognition and are supported, as well as it is transmitted through social organizations in the present day world. But this is not an epistemic claim and does not give any indication about the nature of mathematical knowledge. This claim only states the social situation in which mathematical knowledge, actually any knowledge, is created and transmitted. It says nothing about what mathematical knowledge is.
The two claims in the above mentioned view which indicate something about nature of mathematical knowledge are not tenable. The role of consensus in accepting or rejecting a mathematical claim is not epistemically very important. Consensus in mathematical issues is strictly governed by accepted axioms and logical principles. The consensus is more about whether the principles are correctly applied or not, in other words, to eliminate human error in application of principles.
The other assumption that mathematics is an “experimental inductive science” is plainly wrong. No mathematical statement (proposition, theorem) is accepted to be true or false on the basis of induction. Acceptance always requires deductive proof. Examples from applied mathematics will not work here, for reasons discussed later in this note.
Has this been always the case historically? Do Sulva-sutras use deductive logic to prove their results? Did ancient Egyptians deductively prove their rules of measuring land? Perhaps not. But mathematics has moved far from that position and mathematicians today see the pitfalls in accepting inductive results without support of deductive proof.
To my mind the idea of mathematics as an experimental inductive science arises because of conflating three related but epistemically significantly distinct processes in mathematics: (1) Historical evolution and investigations in mathematics, (2) Pedagogy of mathematics, and (3) justification in mathematics.
Evolution and investigation:
All human knowledge necessarily involves sense impressions at the level of basic concept formation, if they be communicable to others and usable in this world. Observation, organization of observations, hypothesis making, generalization and justification of generalizations form part of the evolutionary process. They are important in mathematics as well. But mathematical concepts have a strong tendency to gradually eliminate physical matter, and therefore observation of the outside world. In a sense mathematics through abstraction brackets out physical substance. Even the idea of number, such a simple thing, which we use today has got nothing to do with physical objects. Therefore, the nature of mathematical objects changes, they are no more concrete objects which could be felt through senses and measured in the same manner as physical objects.
Investigation in mathematics:
Mathematical investigation involves both invention and discovery. When you are conceptualizing a line, an angle, a triangle you may be using observation of physical object and abstracting these ideas from that observation. But once you have defined and fixed them your invention ends, and this invention becomes independent of you. The relationships between the angles, lines and triangle etc. are already determined in defining them. Now you have to discover them. This is the confusion and perplexity Lockhart is talking about when he says “… not because it makes no sense to you, but because you gave it sense and you still don’t understand what your creation is upto”.
Justification
So then, ‘does’ or ‘does not’ investigation in mathematics involve observation, patterns, hypothesis making, generalization, and induction? Yes, it does. Both of physical objects to create abstract notions of mathematical objects, and then of mathematical objects to arrive at propositions. But the observation, hypothesis making and generalization of mathematical objects is quite different in nature than those of physical objects. In physical objects you observe with sense, you observe their physical proprieties, you generalize about their behaviour, and postulate abstract notions (example force, friction, magnetism) to explain the behaviour of physical objects. In mathematics you notice patterns of logical conceptual relationships only through your mind, and look for conceptual relationships between mathematical objects. In physical objects their relationships have to be discovered through the observation of world outside, and the relationships are contingent, inductively established, and imagination of different relationships remains possible. In mathematics no outside observation is needed, they emerge out of the concepts themselves, and are therefore necessary. Using the idea of observation, patterns, hypothesis, generalization etc as if they are the same in mathematical and physical world is mistaken. Calling mathematics an experimental inductive science is like calling “mathematical induction” an inductive process in the same manner as we use induction in sciences. Therefore, justification of mathematical statements which are about relationships between abstract objects and can never be captured by senses, is impossible through experimental and inductive processes. But fortunately is also unnecessary. Because the deductive logic itself is enough to establish such relationships. The process of justification is internal to mathematical systems, while this is external to theories of physics. In its evolution mathematics has to breakout of the limitations of the physical, while physics has to again and again ground itself in the physical reality, even when it often dares to take flights with mathematics. Mathematics has to cross the limits of induction to realize itself, while physics has to worship induction to retain its worth. That is why physics can say something substantial and useful about the world as it is, and mathematics on its own is incapable of any knowledge of the physical world.
Pedagogy of mathematics:
Use of concrete objects in concept formation and initial visualization of relationships in primary mathematics is essential. and may be useful throughout mathematics teaching. Observation, hypothesis making, generalizations etc. are all very important abilities and tools in learning and creation of mathematics. But the justification and critique in math breaks free of observation etc. of the physical world. Therefore, the pedagogy of math has to be such that it facilitates this gradual independence from observation and not its opposite. The process of learning and process of criteria for truth on one hand, and process and criteria for justification on the other differ substantially. While in sciences that gap is relatively smaller.
Applied mathematics
Are my claims of mathematics being abstract, deductive etc. true for applied mathematics? Depends how one understand applied mathematics. In my view applied math saddles two horses: pure math and the discipline in which this is being applied. The physicist while using s=ut+1/2vt^2 is using pure math in derivation of the equation, with idealized concepts of s, u, v, and t. These concepts are capable of relating to the real world phenomena, when the physicist supplies particular values for them s/he is using approximations of physics. Also when s/he claims that this equation is applicable in a particular context it is a claim from physics and not from math. For example, this equation does not take into consideration other forces that may be influencing motion of an object in real world environment.
It seems to me curriculum and pedagogy of mathematics cannot be built entirely on the idea that it is an experimental inductive science. But this idea can be used in some situations with caution to go beyond it; as this is also partially true in investigation and historical evolution. However, as said earlier is patently false in justification and therefore fails to capture nature of mathematical knowledge and even in reliable characterization of mathematical procedures.
One may say that this is not how mathematical activity happens in the real world or society, in this note mathematics is being abstracted and one aspect of math is being ballooned ignoring many others. There is time and space constraint here to go into detailed of this. But in short, educational worth of the discipline lies in the specific human capabilities it may develop. The core of mathematics is axiomatic deductive systems. and development of math even historically as well as in an individual mind has a very strong tendency to move in this direction.
Math is useful in other disciplines because this is empties out of all concrete substance, and therefore, giving appropriate meaning to its symbols by different disciplines through conventions and propositions from those disciplines becomes possible. Students’ attention should be drawn to the use of conventions and principles which come from the user discipline, and they should not be dubbed as mathematical.
This view of mathematics is fiercely contested, and one can reject it. But then that person also loses the soul of mathematics. J
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23 December 2018
Posted by rdhankar 
Thinking Aloud 