Philosophy as mathematics with non-mathematical language

By Aristophanes


What is the purpose of philosophy? For those with no formal training in the discipline, academic philosophy can seem obtuse and of little importance. Many often contrast philosophy, and the humanities at large, with the study of the physical sciences and their cognate disciplines, such as engineering and mathematics. While philosophy may be interesting and enjoyable to learn, it can tell us little about the true nature of the world, they say. What use is mere thought if it is not first grounded in empirical observation and experimentation? According to this line of thinking, philosophy, the study of theoretical foundations, is of little use to modern society.

But I think this conception misunderstands the role of philosophy. The discipline should not be thought of as wholly separate from the physical sciences, but rather as complementary to them. Philosophy is not an isolated field; to philosophize, one must philosophize about something. Consequently, philosophy is divided into countless subfields, including the philosophy of mind, philosophy of language, moral philosophy, political philosophy, epistemology, action theory and, yes, philosophy of science. One might even philosophize about philosophy itself, as we are doing here. To philosophize is to search for the deepest understanding of a specific subject, to develop a logical foundation explaining why that subject allows us to further develop our knowledge of the world. Therefore, not only are the physical sciences enhanced by philosophy, they are at their core utterly dependent upon it.

That’s all well and good if we’re merely focused on establishing new fields or explaining the logic of current empirical practices, one might say, but what about philosophy helps us expand our knowledge beyond the development of simple foundations? When we engage in thought experiments, one of the central procedures of philosophic reasoning, aren’t we just discussing matters that do not exist in reality? How on Earth could such a practice tell us anything about the world?

To this I simply reply: what is mathematics but reasoning about matters that do not necessarily exist in reality? And yet mathematics — even theoretical mathematics — is of obvious importance to the continued development of the physical sciences. Philosophy is merely mathematics performed with a different language. Indeed, mathematics is but one system of formal, or symbolic, logical reasoning, whereas philosophic reasoning can be performed either formally or informally. Through the use of proper translation, philosophical arguments can also transform from formal to informal and vice versa.

For example, take the classic syllogism below:

All men are mortal. Socrates is a man. Therefore, Socrates is mortal.

This is an informal logical argument because it is non-symbolic. We can transform this argument into a formal language by replacing the informal language with representative symbols:

“All men are mortal” becomes “All y are z”

“Socrates is a man” becomes “X is y”

“Therefore, Socrates is mortal” becomes “Therefore, X is z”

Our formulation now reads:

All y are z. X is y. Therefore, X is z.

We can further translate this new formal argument into a formal meta-language:

“All y are z” becomes “A”

“X is y” becomes “B”

“X is z” becomes “C”

So our updated formulation now reads:

A. B. Therefore, C.

We can substitute common mathematical operations for “therefore” and the current punctuation:

A + B = C

In philosophy, we can transform an informal argument into a formal argument by substituting symbols for propositions. In mathematics, we can similarly transform a formal argument (or equation) into an equivalent formal argument by substituting variables for numbers:

“6/3 = 2” becomes “x/y = z” 

Argument translations, both in philosophy and in mathematics, are often most useful when some variables are known and others are not. Therefore, a simple mathematical problem may read:

x/2 = 4. Solve for x.

In the above equation, we have just enough information to logically deduce x without reference to anything but mere thought and our knowledge of the meaning of mathematical operations. We merely insert “8” in place of “x,” see that both sides of the equation are equivalent, and understand now that “x = 8.”

In a philosophical argument, we may know that, for example, “Zeus is not mortal” and that “all men are mortal,” but we are not yet given information on what results. However, we can readily deduce what results by thinking of the argument in a mathematical way:

Zeus is not mortal + all men are mortal = x

We solve for x by deducing that if all men are mortal, and Zeus does not have that attribute, then x equals “Zeus is not a man.”

Zeus is not mortal + all men are mortal = Zeus is not a man

Of course, it must be noted that displaying informal philosophic arguments using mathematical operations is not always quite as useful as displaying mathematical arguments using mathematical operations. If an equation read “1 + x = 3,” we would know that x = 2. But if the philosophic argument read “Zeus is not mortal + x = Zeus is not a man,” there are multiple possible meanings of x. For example, x could mean “all men are mortal” or it could mean “No thing named Zeus is a man.” Nevertheless, displaying philosophical arguments using a mathematical formulation remains a useful exercise for illustrating the many similarities between philosophical and mathematical reasoning.

By thinking of philosophy as mathematics with words rather than numbers, we can more easily understand why the discipline of “mere thought” is still useful in expanding our knowledge of the world. ■


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