Take a very large number, say, 1 quintillion. Written out, it looks like this: 1,000,000,000,000,000,000. It can also be expressed as 1018 or 10.E+18.
I doubt that any human being has ever discerned 1 quintillion discrete objects in a single moment. Including the constituents of all of the stars and planets, there may be more than 1 quintillion particles of matter in the visible portion of the sky on a clear night. But no person may reasonably claim to have seen all of those particles of matter as individual objects.
I doubt, further, that any human being has ever discerned 1 million objects in a lifetime, even a very long lifetime. And if I’m wrong about that, it’s certainly possible to conjure a number high enough to be well beyond the experiential capacity of any human being; 101000, for instance.
Despite the impossibility of experiencing 101000 things, it is possible to write the number and to perform mathematical operations which involve the number. So, in some sense, very large numbers “exist.” But they exist only because human beings are capable of thinking of them. They are not “real” in the same way that a sky full of stars and planets is real.
Numbers and mathematics are rational constructs of the minds of human beings. Stars and planets are observed; that is, there is empirical evidence of their existence.
Thus there are two1 types of scientific knowledge: rational2 and empirical. They are related in the following ways:
1. Rational knowledge builds on empirical knowledge. Astronomical observations enabled Copernicus to devise a mathematical heliocentric model of the universe, which was an improvement on the geocentric model.
2. Empirical knowledge builds on rational knowledge. Observations aimed at verifying the heliocentric model led eventually to the discovery that the Sun is not at the center of the universe.
3. Empirical knowledge may affirm or contradict rational knowledge. Einstein’s general theory of relativity, which is given in a paper written in 1915, says that light is deflected (bent) by gravity. Astronomical observations made in 1919 affirmed the effect of gravity on light. Had the observations contradicted the postulated effect, the general theory (if any) might be markedly different than the one set forth in 1915. (A scientific theory is more than a hypothesis; it has been substantiated, though it always remains open to refutation.)
4. Rational knowledge may lead to empirical knowledge. One of the postulates that underlies Einstein’s special theory of relativity is the constancy of the speed of light; that is, the speed of light is independent of the motion of the source or the observer. This is unlike (for example) the speed of a ball that is thrown inside a moving train car, in the direction of the train car’s motion. An observer who is stationary relative to the train car will see the speed of the ball as the sum of (a) its speed relative to the thrower and (b) the speed of the train car relative to the observer. Einstein’s postulate, which drew on James Clerk Maxwell’s empirically based theory of electromagnetism, was subsequently verified experimentally.
These reflections lead me to four conclusions:
- Knowledge is provisional. Human beings often don’t know what to make of the things that they perceive, and what they make of those things is often found to be wrong.
- When it comes to science, rational and empirical knowledge are intertwined, and their effects are cumulative.
- Rational knowledge that can’t be or hasn’t been put to an empirical test is merely a hypothesis. The hypothesis may be correct, but it doesn’t represent knowledge.
- Empirical knowledge necessarily precedes rational knowledge because hypotheses draw on empirical knowledge and must be substantiated by empirical knowledge.3
* * *
Thomas M. Lennon and Shannon Dea, “Continental Rationalism,” Stanford Encyclopedia of Philosophy, April 14, 2012 (substantive revision)
Peter Markie, “Rationalism vs. Empiricism,” Stanford Encyclopedia of Philosophy, March 21, 2013 (substantive revision)
What Is Truth?
Are the Natural Numbers Supernatural?
Pinker Commits Scientism
The Limits of Science (II)
The Pretence of Knowledge
“The Science Is Settled”
The Limits of Science, Illustrated by Scientists
1. This post focuses on scientific knowledge and ignores other phenomena that are sometimes classified as branches of knowledge, such as emotional knowledge.
2. In this context, rational means by virtue of reason, not lucid or sane. The discussion of rational knowledge is restricted to knowledge that derives from and is a logical extension of observed phenomena, as in the example with which the post begins. I will not, in this post, deal with intuition, innate knowledge, or innate concepts, which are also treated under the heading of rational knowledge.
3. Unless it is true that human beings are born with certain kinds of knowledge, or with certain concepts that can be filled in by knowledge. The article by Markie treats these possibilities at some length.