Nagel’s Comparison and Contrasting of Philosophy and Mathematics

Today’s post is one in a series, devoted for now to Thomas Nagel’s What Does It All Mean? A Very Short Introduction to Philosophy,* a series constituting the Introduction to Philosophy Initiative presented to the world in “Announcing the After Aristotle Introduction to Philosophy Initiative.”

There is a “Table of Contents” listing the titles of and links to the posts published in the series, at:

The “Table of Contents” for the “After Aristotle Introduction to Philosophy Initiative”

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One primary question that anyone beginning the study of philosophy should want to have answered is that of just what philosophy is. As I noted in the immediately previous post, “Nagel’s Comparison and Contrasting of Philosophy and Science,” the “Introduction” to What Does It All Mean? offers us three ways of beginning to answer the question. One is that of contrasting the method of philosophy with those of science and mathematics. A second is that of putting before us a set of illustrative problems that fall within the scope of philosophy and that (p. 4) “come up again and again.” The third is that of identifying what he calls (p. 5) the “main concern of philosophy,” which is “to question and understand very common ideas that all of us use every day without thinking about them.”

The immediately previous post just mentioned took up one half of the first way of answering the question, that of Nagel’s comparison and contrasting of the method of philosophy with that of science. The present post takes up the other half, that of his comparison and contrasting of the method of philosophy with that of mathematics.

The latter comparison and contrasting is found in the same passage as the former. In it Nagel tells us (p. 4):

Philosophy is different from science and from mathematics. Unlike science it doesn’t rely on experiments or observations, but only on thought. And unlike mathematics, it has no formal methods of proof. It is done just by asking questions, arguing, trying out ideas and thinking of possible arguments against them, and wondering how our concepts really work.

In the previous post, I pointed out that the proposition, “Unlike science it [philosophy] doesn’t rely on experiments or observations, but only on thought,” is a complex proposition and I went on to break it down into some of its logical components. Similarly, the first task of the present post is that of pointing out that the proposition:

And unlike mathematics, it [philosophy] has no formal methods of proof.

is a complex proposition, implying a conjunction of two propositions. One of the component propositions of that conjunction is the proposition that:

Mathematics has formal methods of proof.

The other component proposition is the proposition that:

Philosophy has no formal methods of proof.

In other words: mathematics is, even strictly speaking, a theoretical discipline capable of genuine proofs, that is:

Mathematics is a demonstrative science.

Philosophy is not, however, strictly speaking, a discipline capable of genuine proofs; that is:

Philosophy is not a demonstrative science.

Repeating, very nearly, word for word that which I said in the previous post about Nagel’s proposition that philosophy does not rely on experiments or observations: The proposition that philosophy is not a demonstrative science brings us to the heart of the present post’s matter. It is, of course, a perfectly respectable philosophical thesis. Indeed, many philosophers for whom one should have the utmost respect agree with it; Nagel is just one of them. But yet one can ask whether it is true, for it is not the only logically possible position that one could hold. One could also hold that:

Philosophy is a demonstrative science.

And indeed many philosophers for whom one should have the utmost respect agree with this latter thesis. There are major traditions in the history of philosophy that uphold the thesis that (discontinuing for the moment my nearly word for word repetition) philosophy is a demonstrative science. One among them is the Aristotelian tradition, some or even most of the members of which have held that there is a genuine philosophical proof, for example, of the existence of God.

Let me interject parenthetically that I am not making, here and now, any claim that that argument or any other argument that various members of the Aristotelian or any other philosophical tradition have put forward can stand as a genuine proof or demonstration. Nor am I, here and now, claiming that they cannot. I do, however, promise that, at some point, albeit in the relatively distant future, I will subject that argument in particular and others as well to rigorous examination.

Exiting, then, the parenthetical excursion, I have pointed out that there are two logically possible positions that one can take in answer to the question of whether philosophy is a demonstrative science, the affirmative one that philosophy is a demonstrative science and the negative one, contradicting the affirmative answer and accepted by Nagel, that it is not a demonstrative science.

At least one of the two theses has to be true, and they cannot both be true.

Nagel, however, sets before us the thesis that philosophy is not a demonstrative science as true, without telling us why we should accept it, rather than its opposite, as true. My second critique of Nagel’s introduction to philosophy, then, is similar to my first, advanced in the immediately previous post: Just as he presents us with no argument, no setting forth of any reason why the thesis that philosophy does not rely on experiments or observations should be accepted as true, so he presents us with no argument, no setting forth of any reason why the thesis that philosophy is not a demonstrative science should be accepted as true. He does so even as one who holds that philosophy “is done just by asking questions, arguing, trying out ideas and thinking of possible arguments against them….” Surely, however, if philosophers should be engaged in “arguing” and “thinking of possible arguments against” certain “ideas,” he himself should have devoted some effort to “arguing” and of “thinking of possible arguments against” the thesis that philosophy is a demonstrative science and in favor of his thesis that it is not. He simply asserts that philosophy is not a demonstrative science; he does not show, demonstrate, or prove it.

Where does that leave us now? Well, we can observe that, for Nagel, while philosophy is an argumentative discipline or theoretical activity, it is not a demonstrative science. That observation leaves us with two things that need to be done in posts to come in the relatively near future. One of them is that of spelling out what arguments are, in the relevant sense of “argument,” and what conditions an argument has to meet if it is to be a proof or demonstration; this will constitute one small part of an introduction to logic. The other is that of taking note of some implications that Nagel’s denying that philosophy is a demonstrative science has for the discipline, implications evident in the second chapter of What Does It All Mean?, “How Do We Know Anything?”

Before turning to those tasks, however, we need to take up the second and the third of the three ways, noted above, which the first chapter of What Does It All Mean? offers us of determining what philosophy is. In the next post, we will take note of and reflect on the second of the three ways, that of putting before us a set of illustrative problems that fall within the scope of philosophy. In the post following that, we’ll turn our attention to the third of the three ways, that of identifying what Nagel calls (p. 5) the “main concern of philosophy.”

Until next time.

Richard

*Thomas Nagel, What Does It All Mean? A Very Short Introduction to Philosophy (New York and Oxford: Oxford University Press, 1987). The book is available through, of course, Amazon.com, as well as elsewhere.

 

 

About Rchard E. Hennessey

See above, "About the Author/Editor."
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