It is widely recognized that philosophers get an idea that has some merit—within limits and in some applications—and then they forget about the limits. They imagine their idea as a theory of how the entire universe works.
We can take Plato as an example. He said “The study of mathematics sets into motion a mental organ worth more than a thousand eyes, because by it alone can truth be perceived.” [1] Consider some objects of Plato's mathematics. A plane: like the surface of a flat table—but idealized: perfectly flat, without thickness, and without edges, extending without limit in all its directions. A line: like the line you could draw with a straight edge on a paper on a table top—but idealized: without thickness, perfectly straight, and extending without limit in both of its directions (for example, to the right and left). A point: like the dot you could put on a paper with a pencil—but idealized: without any real size, just a single location.
Suppose we asked the question, “Are the theorems of geometry true?” Plato and I would agree that they are [2], but what are they true about? Not the molecules of pencil lead on a piece of paper. Approximately true, yes. But the theorems of geometry are not perfectly true about anything in the physical world. They are true only in the ideal world of points, line and planes that we talked about in the last paragraph. So you see that here, in this application, truth (perfect truth) exists only in the ideal world, and not in the physical world. Now guess what this insight leads to in the hands of a philosopher: a whole philosophy of truth and idealism. While mathematics has tremendous application to the physical world, we must beware of philosophical generalizations.
Here is what Hugh Ross has to say about Plato, Aristotle, and astronomy:
Experimental science was considered 'mere engineering'—not worth the attention of great intellects. Plato wrote in the republic: 'It is by means of problems then, said I, as in the study of geometry, that we will pursue the study of astronomy too, and we will let be [ignore measurement of] the things in the heavens, if we are to have a part in the true science of astronomy.' Aristotle, one of Plato's students, philosophized that any past motion of the earth must naturally be toward the center of the universe. Therefore, he said, it is clear that the earth does not move. [3]Ross points out that the Greek observational astronomers had previously found out the relative distances between the earth, moon, and sun, their sizes, and the relative remoteness of the stars—and had concluded that the earth revolved around the sun.[4] For the world to learn about their findings, it had to wait for the invention of the printing press.
The story continues with Aristotle's philosophy being integrated with church teaching by Aquinas, the rediscovery of the Greek astronomers, the resistance to Galileo by those he called "academic philosophers", and more of his story. (to be continued)
1. See Marvin Greenberg, Euclidean and Non-Euclidean Geometries, W. H. Freeman, San Francisco, 1980 (return to text)
2. The discovery of non-Euclidean geometry has shown that there are ideal worlds other than the one Plato had in mind. It is true that one of these other ideal worlds is a better model of the physical world than is Euclidean geometry. But this has nothing to say against the fact that the theorems of Euclidean geometry are true about the figures in Euclidean geometry in the space of Euclidean geometry.(return to text)
3. Hugh Ross, The Fingerprint of God, page 14. My remark in brackets.(return to text)
4. ibid, page 14.(return to text)