My point in this entry is to argue for the existence of points, a controversial topic in metaphysics. To put the argument very simplistically, if we can agree that geometry and calculus tell us what exists, then a straightforward examination of these theories reveals that they imply:
- There exists a midpoint of any line segment.
- For each real number x, there exists a point on the mathematical line that is a distance x from the origin.
Someone might ask, “Given what we know about points how do we go about detecting them?” My response is, ”Given what we know about points, we should not be trying to detect them.”
Mathematicians justify their statements by proving them, by deducing them from the axioms, but this remark de-emphasizes the fact that the axioms themselves need justification. What axiomatization does is systematize claims, not justify them.
When it comes to justifying the ascription of “truth” to mathematical existence claims, we should consider mathematics to be part of science, not a parallel discipline to science. All true mathematical claims should be justified the same way other scientific claims are—by their empirical success, by how they fit into a larger network of claims that is also justified by its success. But mathematics is a very special part of science since its claims, and those of formal logic, are much less impacted by new empirical evidence.
Mathematics demonstrates its empirical success because very often when the principles of mathematics are violated in our scientific reasoning the probability soars that the bridges we design will fall down and that absurdities will be deduced.
Let’s turn now from mathematical points to physical points, points of space, of time and of spacetime. Our well accepted physical theories imply that
- The path that Achilles takes from this point to that point has a midpoint.
- There is an instant, a point of time, when that uranium nucleus emitted a neutron.
If these statements are true, then non-mathematical points exist. There are no good reasons to say these are mere approximations to the way things are. There are many approximations in science—a molecule is approximately a point particle—but points themselves do not lose their ontological standing simply because molecules are not really point particles.
Nor is there an unsolvable problem of epistemic access to points, of how we know about points. We made up the theory of the points, and that’s how we know about them.
We justify points holistically by appealing to how they contribute to scientific success, that is, to how the points give our science extra power to explain, describe, predict, and enrich our understanding. But we also need confidence that our science would lose too many of these virtues without the points.
We should reject the various versions of skepticism about points, such as
(a) conventionalism; according to which there may be other undiscovered and equally adequate mathematical systems that make no use of points;
(b) semantic instrumentalism, according to which theoretical terms such as “point” are not to be interpreted as referring to anything,
(c) theoretical reductionism, according to which the term “point” is a disguised way of referring to observable phenomena,
(d) constructive empiricism, according to which points may exist, but we are only justified in accepting scientific theories that refer to unobservable entities as "empirically adequate."
Let’s bet on this success. Let’s bet that the truth of point talk and the other talk with theoretical terms is integral to explaining science’s success at making predictions and producing explanations. And bet that the existence of points comes along with the truth of point talk. Point talk is not an idle or extraneous part of science, although we should agree with Kitcher that no “sensible realist should ever want to assert that the idle parts of an individual practice, past or present, are justified by the success of the whole.” We should not insist, though, that the successful reference of “point” is a necessary condition for the success of theories that incorporate the word “point.” And even though some theoretical terms of our best contemporary science will be regarded as non-referring by future generations of scientists, there is no good reason to bet that the term “point” will be one of those terms.
One last comment on the holism involved in justification. We are justified in adding points into our ontology because they are indispensable to the rest of the package that we have good reason to accept as approximately true. This package is large. In contains the lack of sufficient reasons to doubt that motion is continuous rather than discrete, the need in so many places to use the principles of geometry, calculus and logic, the need to embrace the general theory of relativity which describes the details of all motion in a background spacetime composed of points that are indiscernible one from another, the belief that quantum mechanics is approximately true and that space and time are not quantized in quantum mechanics, the recognition that the sciences have made so many varied, successful predictions, the presumption of the overall instrumental success of scientific methods across the history of science, and the assumption that we are not dreaming.
That is the point of a point.