Monday, April 11, 2016

What's the point of a point?

I don’t know whether it is more shocking to common sense to be told there are zero-dimensional mathematical points or to be told water is mostly oxygen. Both illustrate how common sense often has to take a back seat when it conflicts with progress.

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. 
If these two statements are true, then points exist. How do we know the two statements are true? The answer is rather complicated. Part of the answer is to show that the statements are not like the statement that there exist horns on unicorns.

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."

By contrast, we should embrace this quotation from Putnam: “The positive argument for realism is that it is the only philosophy that doesn't make the success of science a miracle.”

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.

Brad Dowden
Department of Philosophy
Sacramento State


  1. Brad, I have a few questions.

    1. Except for referring to points as zero-dimensional, you don't really give us a reason to be dubious of the existence of points to begin with. What do you think is the strongest basis for point skepticism?

    2. Previous to your post, Tom wrote a post defending the existence of holes. His argument in the post assumed that holes supervene on matter, but in the comment section he allows the existence of holes in abstract objects. Do you think holes and points are similar in regard to their ontological status, or is their a principled basis for being a point realist and a hole skeptic?

    3. When I think about a proposition like "Every line segment has a midpoint" I find myself thinking it just means that every line segment can be divided into two line segments of equal length." It seems like the other propositions you cite may be susceptible to similar procedural analyses. Is there a clear counterexample to or a disproof of this notion?

  2. Randy, you ask interesting questions. I’ll reply to them in order.

    (1) I think one strong basis for point skepticism is Bas van Fraassen’s agnosticisim, which implies acceptance of the best scientific theories requires only believing that the theories adequately describe the observable world and not that they are true. My argument for points depends heavily on not being agnostic about our best scientific theories. I believe in molecules and species; he’s agnostic about them.

    (2) Holes and points have some similarity. However, Tom argues informally, like Aristotle. My way of arguing demands that in scientific theories there be truths having the form of an existential quantification, in which we quantify over points. I haven’t been convinced that scientific theories need quantification over holes, only that informal reasoning finds hole talk occasionally useful.

    (3) You can’t escape points this way. The definition of the line, as the mathematical continuum, requires there to be points throughout the line. Points of space have an intimate connection to the continuity of space. Intuitively, continuous change and continuous objects are smooth and gap-free. The more technical idea, which took hundreds of years to define rigorously, is that something continuous is composed of very densely packed points that are so close together that no point has a next point, and yet between any two points there are as many other points as there are real numbers. So, points need to exist in order for us to have a fruitful way of explaining continuity rigorously. Continuity is needed to escape Zeno’s Paradoxes of motion. We say Achilles’ path as he chases the tortoise is a continuous function of time. A consequence is that Achilles passes by an infinite number of physical point-places in a finite time. We are justified in saying physical space is continuous, in the technical sense, for two additional reasons: (1) It is too difficult to use theories that imply there are chunks of space with no parts. (2) We have no reason to doubt that when mice and molecules move, they move continuously through space. Nobody has noticed instantaneous jumps to new locations, even within an atom.

  3. Brad, this is very interesting. I was not aware that the only live account of the continuum depends on the notion of a point. I would have guessed that the argument would have been more like: Given our commitment to points, space must be dense with them in this sense. Intuitively, the assumption of continuity seems to militate against the existence of points.

  4. Randy, Your intuition that a continuum is too smooth to be composed of indivisible points is shared by a great many people. Many philosophers, but fewer each year, object to a continuum being constructed from points. The Austrian philosopher Franz Brentano believed with Aristotle that scientific theories should be literal descriptions of reality, as opposed to today’s more popular view that theories are idealizations or approximations of reality. Continuity is something given in perception, said Brentano, and not in a mathematical construction; therefore, mathematics misrepresents. In a 1905 letter to Husserl, he said, “I regard it as absurd to interpret a continuum as a set of points.” But the contemporary definition of the continuum needs to be thought of as a package to be evaluated in terms of all of its costs and benefits. From this perspective the point-set analysis of continua has withstood the criticism and demonstrated its value in mathematics and mathematical physics.

  5. Brad, thanks for that. In your post when you wrote "There are no good reasons to say these are mere approximations to the way things are," I took you to mean that we have good reason to accept points as literally existing in reality. Is that wrong? If it is right, then are you saying that you do not subscribe to what you call above "the more popular view that theories are idealizations or approximations of reality?" If you do reject it, then would that not entail that you actually accept the Brentano/Aristotle criterion, but reject Brentano's view that points are inadequate in this sense?

    1. Randy, It is right that we have good reason to accept points as literally existing. I also believe that theories try to capture reality but often have to settle for being idealizations. However, I don't think the existence of points is such an idealization, nor is the existence of stars and cows and DNA. Are cows mere idealizations or approximations?

    2. Brad, I'm willing to bet that our concept of cow, star, and DNA will all be considered crude approximations from the perspective of the science of 1,000 years from now, at least insofar as they are properly analyzed into constituent concepts that we now understand quite poorly. Woudn't you bet that, too? I also wouldn't want to bet much against the development of alternative geometries that don't employ the concept of a point being developed within that period, either. I admit I can't imagine what that even means. But it seems like Kant couldn't imagine what 4-dimensional space would mean either.

      My only question above though was just whether you don't in fact accept Brentano's expectation that science be literal descriptions of reality. (Because it initially read to me as if you found it problematic.) In that case you just must be replying that, contrary to Brentano's intuitions, the continuum literally is dense with points.

    3. Randy, you and I will bet differently on what is going to be called “crude.” Would you bet that the current scientific belief that the Earth isn’t flat will someday be called “crude”? I wouldn’t. Your point about scientific change is very attractive to many philosophers of science, but not to me. The point, when developed a bit, goes by the name of the “pessimistic meta-induction.” You know about this, but maybe our readers do not. It says, if we consider all the abandonment of scientific theories that has taken place over the centuries, then we ought to expect that our current best scientific theories will themselves be abandoned, and hence that we ought not to assent to them now. I am an opponent of this meta-induction if it is supposed to imply that it is rational now to disbelieve everything science tells us about what exists. Even if we reject some theory in the future, about, say, the physiology of cows, I am betting this rejection won’t be replaced by some new theory that requires us to believe that back in 2016 there were no cows. I’m open to the slim possibility that right now there are no cows, and no points, but I wouldn’t bet against them. However, I adopt ontological views in degrees. I hold more strongly to belief in cows than in points.

      You also said you “wouldn't want to bet much against the development of alternative geometries that don't employ the concept of a point” in the future. We have those alternative geometries now. However, they aren’t useful for science; the current geometries are.

      Regarding your not being able to imagine these alternative geometries, it is always great when we can imagine what it would be like if this or that were so, but successful imagination is not a requirement on whether to accept or reject a theory. None of the experts in quantum mechanics is able to imagine quantum mechanical phenomena, but they are able to use the theory to explain and predict all sorts of other observable phenomena, and this theory is the best of all current scientific theories.

      Now, about your comment on Brentano. I’m not much of a Brentano scholar, but I think we understand him differently. I agree with him that science tells us about reality, but I disagree with him that science should be literal in the sense of avoiding unobservables. The Logical Positivists who came after Brentano developed Brentano’s idea here by demanding that unobservables be definable in terms of observables, or else talk of unobservables is cognitive nonsense. Thankfully, few people today are doing metaphysics in the style of the Logical Positivists.

    4. Brad, thanks for all of that. Very interesting. I don't understand Brentano very well at all, and wasn't aware of his stance on unobservables.

      I actually agree much more with you than I disagree, especially about degrees. My bet on cows and DNA is not so much that people in the future will come to see them as we see witches or caloric or gravity now, i.e., as entities or forces we thought existed but just plain don’t. Rather, I suspect that we may come to see them as useful at a particular level of understanding but which don’t ultimately succeed in carving nature at its joints.

      I agree with you that it is irrational to adopt a perspective from which the success of science is miraculous. For me this means that we should strive for some coherent and non vacuous way to express our basic hunch that the reason science is succeeding is that our theories are getting to be more accurate representations of reality. Right now I think we are still trying to develop the conceptual resources we need to express this in a way that holds up to analysis. I think we are largely still mired in what Wittgenstein called a picture theory of language which we inherited from Locke and Descartes, and as a result we don’t really know what we are saying when we say that our theories are approximately true or closer to the truth. The easiest stance to adopt is van Fraassen’s that we are getting better at prediction, and just remain agnostic about the question why, but that is deeply dissatisfying and I think defeatist.

      As far as the pessimistic induction is concerned, it’s possible that I take it more seriously than you do (though I don’t find anything particularly pessimistic about it). The reason I think the pessimistic induction is partly correct, though, is not so much that we can reasonably infer from the past that science will constantly be massively revised in a Kuhnian sense. Rather it is that we have really strong theoretical reasons for thinking this may happen in the near future.

      The reason is that we seem to be at the brink (say, within 100 years, maybe a lot sooner) of creating superintelligent machines. We don’t really know what these are going to look like, but we know they will work very differently than human brains. Even if these machines are essentially human brain simulations, they will work at speeds that are millions of times faster than human brains. If they are not whole brain simulations, then they will probably have dedicated perceptual modules for doing what humans now do through laborious conscious calculation. To me, it is just not reasonable to bet that such intelligences will be restricted to the same basic ontologies and fundamental concepts that humans use today. So much of the way we think, as well as the way we conduct inquiry is the result of limitations that superintelligent agents won’t share.

      I think from the point of view of superintelligent agents, an Einstein will be a bit like a chimp is to us. We know we are vastly more intelligent than chimps, but in becoming so we have lost the ability to think like one. Chimps don’t really understand that we are smarter than them, but we will understand that superintelligent machines are smarter than us. If they are doing all the successful science in the near future (assuming they choose to keep us around) we will credit them with the real understanding of the world, even if we can’t understand anymore how they are figuring things out.

    5. Randy, I agree with you about the superintelligent agents of the future, that they probably won’t think like us, and will think better and faster than us, and that we, if entities like us are still around at the time, will appreciate that they have these abilities. I am not positive though that they will be biologically completely unlike us. Because I think functionalism is basically an incorrect philosophy of mind since it leaves out biology (here I am adopting John Searle’s position over Daniel Dennett’s), I think these superintelligent agents will have some sort of biology, and I think it is a serious possibility that agents will have some parts of our DNA; it is possible that your descendants and my descendants will become those superintelligent agents—that humans will merge with machines. If we don’t merge, then I am afraid that humans will simply be conquered and replaced by superintelligent machines since I don’t believe that being smart implies being nice. Human civilization will come to an end, though perhaps some of us will live on to inhabit the cage next to the chimpanzee cage in a future zoo.

      You made another point. You recommend that we adopt an anti-realist position about science, such as van Fraassen’s constructive empiricism. Your main reason is that we realists cannot clear up the vagueness in our idea that science is approximately true, and our idea that scientific progress brings us closer to the truth. I agree with you that it’s extremely difficult to be more precise, but I don’t think this is a sufficient reason to reject realism.

    6. Brad, thanks, I see that besides thinking very similarly, we share common assumptions about likely possible futures. Though I don't have as well formulated views about the correct theory of mind. I guess the proof will be in the pudding. Searle's argument won't be very impressive to me if it doesn't end up putting some kind of empirical constraint on what can be accomplished within a functionalist AI paradigm.

      I totally agree with you about van Fraassen and realism. I didn't mean to recommend it so much as explain its appeal.

    7. Randy,
      Searle's main point in his theory of mind is that the functionalist theory is incorrect because it doesn't put any empirical constraints on what can be accomplished. It pays no attention to any need for biology, yet from everything we know so far about entities that are intelligent, some sort of physiology (not necessarily human) is required for intelligence, and there is no indication it can be produced merely by clever syntax or symbol shuffling in a computer program. Searle does believe that a machine can be intelligent because he believes we humans are machines.

    8. Brad, that sounds familiar. I think my view is that if the functionalist theory is incorrect then there will almost certainly be forms of cognition and behavior that it just can't produce. Does Searle predict this, or does he just insist that regardless what sorts of behaviors it produces, it can't be intelligent because it's not biological? In my view you need to have some sort of agreed upon operationalization to end a dispute of this kind, and I've never understood how Searle would operationalize intelligence. I just know that he rejects Turing's proposal.

  6. Thank you for your post Professor Dowden, also for the time you have further spent in the comments.

    In your discussion of points I notice that it has come up that the current definition of the continuum relies on points expressed by the real numbers and the real number line. Recently, in Phil 160 we have been exposed to the hyper-real line, which is an even denser continuum than the real line. Would you say that only certain types of points exist (namely points described by/describing real numbers), or that even these weirder points exist? Or perhaps, is this a situation where these types of points exist only as far as they are incorporated into a model that requires them to describe reality?

  7. Stan, interesting question! The hyper-real line implies that the smooth continuum of the real line is not smooth, but is gappy. It implies that between any two real numbers there are hyperreal numbers, and ditto for numbered points. For instance, between zero and any positive real number, there is zero plus an infinitesimal, and zero plus seven times an infinitesimal, and so forth. Because of my being in agreement with W.V.O. Quine about how to tell what mathematical objects exist, I'd say ordinary points exist, but we shouldn't say hyper-real points exist until we can show that they are needed in science. So far, they aren't. The hyper-real points do exist in the more liberal (what I'd say is too liberal) sense of mathematical existence that requires only logical consistency with the rest of mathematics.