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Better Than Pie Charts:

Mathematics as the Ultimate Conceptual Aid, Part Two

David Knott

Might there be alternatives to mathematics?

In order to explain the power of mathematics, I have called attention to the diversity of functions, the high level conceptual manipulation allowed once a sufficient amount of proof has been worked out, and the further expediency granted by notation. And to convince the reader of the discipline’s necessity, I have argued for the glaring fallibility of evolutionarily endowed human intuition, and the ability of mathematics to overcome such difficulties. The efficacy of mathematics is, when the previously observed facts are considered, not so mysterious after all. But having satisfied some of our curiosity on this question, we are faced with different problem. Might it be possible to find a system which possesses the essential attributes of math, but which nonetheless must be considered a different entity? Are there other explanatory, analogical techniques lurking beneath our conscious observation, just waiting to be exploited in describing physical phenomena?

To answer this question, we should consider what sort of characteristics an alternative system must possess in order to perform the same work as mathematics. There must be a high level of representational diversity; there must be a framework within which elements of the system can be freely manipulated; there must be a capacity for incremental progression similar to proof; there must be rigor, consistency, and the simultaneous exploitation of generality and specificity. There must be a universal consensus on standards of notation (assuming there are any) and technique. And there must be a way to convert and deconvert between the system’s representational elements and the physical phenomenon being described. What does this imply about our potential alternative system?

It is possible to imagine a sort of cousin to mathematics that uses different words, different symbols, and perhaps slightly different axioms as starting points. It might, in some instances, be possible to make axiomatic what is in standard math a theorem, and to make derivative what is normally assumed. But despite differing superficially from the standard set of mathematical tools, we would unhesitatingly still call such a system mathematical. And clearly, we do not change the underlying structure of the system by using Chinese characters in place of Roman and Greek ones. To make a potential analogical system non-mathematical, we have to alter a more fundamental principle than notation. How far can we push the envelope? Consider the following bit of argument:

If X, then Y.
X.
Therefore Y.

It strikes me that, without the ability to take a general rule (if X, then Y) and apply it to a particular situation (X), and derive a conclusion (therefore Y), we have absolutely no computational or conceptual power whatsoever, and our system is frozen in place. Statements of the form “if, then” must be functional if we are to make any headway in understanding the world. Consider, for instance, how many problems begin with something like, “suppose there is an electron in a potential well,” or “suppose a volume of oxygen is heated.” These questions that ask you to "suppose" something require the use of an if-then statement that can accept a premise as input. Such machinery comes from the rudiments of logic itself; therefore it may be reasonable to say that our system must be, in some sense, grounded in logic. This is our first requirement for mathematical competitors. What other requirements might there be?

The reader may have had the unfortunate experience of being subjected to a never ending marathon of “how do you know?” type questions by a child (or a sarcastic adult).

Why do you drive a car?
Because I need to get to work on time.
Why?
Because if I don’t, I’ll get fired.
Why would that matter?
Because I need food.
Why?'
Because humans need food, and I’m a human.
Why?
Because of energy conservation laws.
Why do those exist?
Because I said so, now leave me alone.

Anyone who has undergone this interrogation knows that, at a sufficiently fundamental level, one has to make a few assumptions in order to get anywhere. This is what mathematics does in using a set of axioms as a starting point. A particular statement is taken to be self evidently true, and further work consists of deriving consequent facts from the axioms initially chosen. The advantage of this methodology is obvious: if we did not allow ourselves this luxury of assumption, then no work could ever be performed, and a serious body of theory could not develop. But it would be incorrect to think that the axioms of traditional mathematics are entirely arbitrary; there are standards against which we can judge a given axiom. If, for example, we develop a system of reasoning from a particular set of axioms S, and after twenty years of argument and proof, we derive theorems that can be applied in the real world, then in some sense, S becomes testable. If the high level theorems derived from S have no usefulness in describing the natural world, and only confusion and nonsense are produced from them, then we can reasonably believe that our axioms were incorrect. Similarly, if a set of axioms is chosen, but no interesting conclusions can be drawn from it, then we can safely choose another set. We can therefore say that any potential competitor with mathematics must, in order to get off the ground, begin with a set of axioms, and in order to have any utility, must begin with the right set of axioms.

It seems likely, then, that any explanatory framework that might serve as an alternative to mathematics would have to be grounded in logic and possess a set of axioms that leads to useful and rich theory. The theorems produced by our choice of axioms must be diverse enough to encompass the breadth of behavior exhibited by physical objects, and somehow capable of mimicking real world behavior. In addition to these stipulations, quantity and shape must almost certainly be involved if we are to make any serious attempt at representing the physical world, for the mind struggles to formulate any coherent description without using these concepts. Our mathematical alternative must therefore be logical, axiomatic, and concerned with number and shape. We might, upon further consideration, come up with other requirements, but these criteria alone seem to uniquely specify mathematics as the one system successful in filling the role of mind extending, reality representing analogical device. Any axiomatic system, based on logic, and concerned with quantity and shape, rather than displacing mathematics, would probably be classified as mathematical itself. It is fair to say that the king consumes his potential usurpers.

Conclusion

Mathematics is the only game in town. Capable of representing virtually anything, it creeps up in every scientific endeavor in which human beings find themselves engaged. Its uncanny correspondence with reality creates a parallel world in which a person can explore labyrinths of conceptual argument, and perhaps even annex territory for their own delight, fame and intellectual satisfaction. It is unclear what history, what language, and what technology people will study in educational institutions of the future. But there is no doubt that mathematics has a permanent residence on university curriculums: the math that people learn one thousand years from now will be the same math we study today. There is great beauty in the permanence of math, and we have seen here some of the reasons behind that permanence. Whatever else might change, mathematics will guide our species to its next destination.

Alumni Liaison

BSEE 2004, current Ph.D. student researching signal and image processing.

Landis Huffman