“I am a passionate scientist who is passionate about science, but I also think scientism is a huge mistake”, writes Adam Frank, an astrophysicist at the University of Rochester, in an article in *The Big Think*. As another astrophysicist, who has called this blog “defending scientism”, I am inspired to reply.

Such disputes can boil down to what one means by the word “scientism”. Professor Frank quotes one definition as “the view that science is the best or only objective means by which society should determine normative and epistemological values”. On that definition I also would reject scientism (indeed I don’t think that anyone does advocate that position). Science cannot prescribe values or aims. Science is descriptive, not prescriptive, it gives you knowledge, not normativity (instead, values, aims and normativity can only come from humans).

But Frank also expounds:

In the philosophy that would come to underpin scientism, “objective” came to mean something more like “the world without us.” In this view, science was a means of gaining access to a perfectly objective world that had nothing to do with humans. It gave us a “God’s eye view” or a “perspective-less perspective.” Science, according to this philosophy, revealed to us the “real world,” which was the world independent of us. Therefore, its truths were “deeper” than others, and all aspects of our experience must, eventually, reduce down to the truths that science reveals. This is scientism.

I’m close to agreement with this version of scientism. Science does indeed attempt to reveal a real, objective world that is independent of us, and to a large measure it succeeds (though we can never attain any absolute certainty). And yes, science does give us truths about the world (as best as we humans can access them) that are more reliable and more extensive and thus “deeper” than other attempts to describe the world. But no, it is not true that “all aspects of our experience must, eventually, reduce down to the truths that science reveals”. Science is solely about attaining the best and truest description of the world (which includes ourselves) that we can. It doesn’t even pretend to encompass “aspects of our experience” other than that (indeed I’m not even sure what this claim would mean, and, again, I don’t think this is a view that anyone actually holds).

Professor Frank’s objection to scientism is that:

[Scientism] is just metaphysics, and there are lots and lots of metaphysical beliefs […] that you can adopt about reality and science depending on your inclinations. […] Scientism claims to be the only philosophy that can speak for science, but that is simply not the case. There are lots of philosophies of science out there.

So, according to Frank, scientism is just metaphysics, there is no evidence for it, and so adopting it comes down to personal choice, the very opposite of science. Effectively, science does not point to scientism.

I don’t find this critique convincing. In essence, “scientism” is a unity-of-knowledge thesis that the real world is a seamless, self-consistent whole, and thus that the best description of it will also be (or should, at least, aim towards being) a seamless, self-consistent whole. That is, there should be a “consilience”, or self-consistent meshing of different areas of knowledge and of ways of attaining that knowledge. Scientism is a rejection of the idea that there are distinct and different “ways of knowing”, each appropriate to different and distinct “domains” of knowledge.

But is that claim only a metaphysical belief, whose adoption is merely a “faith”? I submit that, no it is not. Instead, it’s the product of doing science and seeing what works best. Science rejects the supernatural, not as an *a priori* commitment, but because models of reality work better without it. In medieval times “the heavens” and the earthly world were regarded as different domains to which different rules applied, but then Newton invented a law of gravity that applied both to the Earth-bound fall of an apple and to the orbit of the Moon, unifying the two domains. Even then, the worldview of a scientist could involve a God (invoked by Newton, for example, to keep planetary orbits stable), but, as science progressed, it was found that we “had no need of that hypothesis“. And it had been thought that living animals were utterly distinct from inanimate matter, but nowadays the disciplines of physics and chemistry transition seamlessly through “biochemistry” into the disciplines of biology and ecology. And any proper account of sociology needs to mesh with evolutionary psychology.

Through that progression we have found no sharp divides, no deep epistemological ravines that science cannot cross. The strategy of unifying different areas of knowledge has always proven the more successful.

Thus science *does indeed* point to scientism, and the unity-of-knowledge view is a *product* of science. It is not simply one of a number of un-evidenced metaphysical possibilities, it is the one to which the history and current trajectory of science points.

And the idea is refutable. If you want to reject scientism then put forward an alternative “way of knowing” that gives reliable knowledge about the world and yet is clearly distinct from the methods of science. Be sure to include a demonstration that such knowledge is indeed valid and reliable, and thus comparable to scientific knowledge, but without using science in that demonstration.

RobWhat you say seems correct to me, Coel.

There are no “alternative ways of knowing” anything. If it’s not knowable by science then it’s simply not knowable. Some seem to want to denigrate scientism in the same way that atheism is denigrated. There seems to be a smear campaign that pushes the idea that scientism is an impoverished world view. But those who denigrate scientism have nothing better than science and cannot tell us anything about these other ways of knowing or demonstrate the sort of knowledge they provide.

I agree that there is only one domain of knowledge and science is how we explore that domain.

RobCoel, I am not a scientist or mathematician but people often point to pure mathematics as an example of a self-contained domain of knowledge that does not depend on observation of the physical world or the methods of science and, thus, as a refutation of scientism. I don’t know enough science or philosophy to answer that but it seems to me that mathematics and science are so intimately entwined as to be one. Can you give us your thoughts on this, Coel?

CoelPost authorHi Rob,

Yes, I should write more on this (I see you’ve found an earlier article). In short, I see maths as having been adopted as a real-world model in the same way that (the rest of?) science was adopted as a real-world model. Thus, the counting numbers and 1+1=2 were arrived at for counting the goats in one’s flock and similar real-world tasks. From there, mathematics got distilled into axioms, and mathematicians build wonderful edifices on those axioms, but the whole thing is still ultimately a real-world model, adopted because it models the world. (Even if not every part of maths has an immediate real-world counterpart, but then nor does every part of physics; one can conceive of, for example, possible planetary systems that are compatible with physics, even if that particular possibility is not extant.)

RobSorry, Coel. I see now that you have deal with this already in your post, https://coelsblog.wordpress.com/2014/05/22/defending-scientism-mathematics-is-a-part-of-science/ which I found a satisfying read.

RogerRob, that is mathematics as seen by a physicist. A physicist might see math as just what empirically works with numbers.

Most mathematicians do not see it that way. They prove things to be true.

CoelPost authorHi Roger, by “prove things to be true” you mean one proves them to be consistent with axioms. But then theoretical physicists might also work out implications of physical models, even if not every implication can then be compared to a real observed entity.

It’s also worth noting (given Godel) that mathematicians cannot prove the self-consistency of their edifice, so in that sense they cannot “prove it is true” (since they cannot prove it is self-consistent). The ultimate validity of the axiom set is then that it works in the real world. (With limited and debated instances where it doesn’t, such as Banach/Tarski and the Axiom of Choice applied to infinite sets.)

I think one can push further, and say that the very meaning of mathematics depends on and derives from it being a real-world model. Without an “interpretation”, maths would just be squiggles on a page without meaning, and the only way of it gaining any meaning is by reference to the real world. Thus Peano’s axioms are only meaningful by relation to real-world concepts, and thus ultimately by reference to counting the goats in a flock.

Put another way, we humans could not do maths except using brains that are operating on real-world concepts.

SchlaflyCoel, this Godel argument is a red herring. It is true that proving the self-consistency of axioms from those same axioms does not really mean self-consistency. It means the opposite. Godel proved that. But Godel would not agree that this stops us from getting mathematical truths.

CoelPost authorI would agree that mathematicians can indeed prove things. I’m only arguing against mathematics being a stand-alone edifice in its own right, fully distinct from the physical world.

RobThanks, Roger. As mentioned, I’m not a mathematician, scientist or philosopher but, from what I’ve read, there doesn’t seem to be agreement, even among mathematicians, about whether mathematics is science.

You say that mathematicians “prove” things to be true. What do you mean by “prove’? Do you mean that they show definitively that a proof is a logical necessity derivable from the rules of mathematics alone and not dependent on observation of the physical world, and that this is in contrast with science which only ever gives us provisional answers that may, with further research, be shown to be incomplete or wrong? If so, do you take this as showing that mathematics is separate from science and that this demonstrates that scientism, as expounded by Coel, is an incorrect world view?

CoelPost authorHi Rob,

Yes, that’s what mathematicians mean by “prove”. Though Godel showed that they cannot show that their axioms+proof are self-consistent, so they either have to just assume that the set of axioms (“rules of mathematics”) that they are using is self-consistent, or they could validate the ensemble by noting that it works in the real world.

As in my other reply to Roger, I think one can go further and say that the very meaning of mathematics has to rest on concepts that we can only obtain as a real-world model. One can only state the axioms of mathematics by using a human language, and human-language concepts are arrived at as real-world models. How else could we develop concepts?

SchlaflyRob, I think that there is broad agreement, among mathematicians, that mathematics is NOT a science. Among non-mathematicians, the differences are not so obvious, as math and science papers can look superficially similar.

Yes, mathematicians prove statements are true, without any need for real-world observations or unusual assumptions.

RobThanks, Schlafly.

I’m not opposed to it in principle, but the “Platonic realm” idea of mathematics seems to me to be at odds with everything else we know about the world and how we come to know it. It sounds like a superstition akin to the spirit world or the supernatural world. There seems to be no evidence or need for it. Its proponents can only point to the fact that we can deduce that 1+1=2 without recourse to empirical verification. But Coel’s explanation of this phenomenon rings truer to this layman than the idea that we somehow have access to some mysterious Platonic realm. That just seems like another hypothesis of which we have no need. Like all knowledge, mathematics must have had it’s origins in and been built on our interaction with the real physical world of which we are a part. That seems like the most parsimonious explanation.

CoelPost authorJust a comment, one cannot deduce 1+1=2 (or any other mathematical truth)

ex nihilo, one can only deduce 1+1=2 from axioms, where those axioms were adopted specifically to lead to 1+1=2 (that is, they tautologically entail 1+1=2).RobI should have added that if what I said above is true, then mathematics would seem to be a part of the rest of science and there seems to be no need to posit a separate realm of knowledge. I guess I just don’t understand why some philosophers and mathematicians seem to want mathematics to be not science. To the layman it seems like just an attempt to protect territory.

CoelPost authorI can fully understand why mathematicians feel that maths is distinct from science (and I don’t think it’s protecting territory). It really is very different in

style. Mathematicians are using reason (with no direct link to empiricism or data), and arriving at truths that seem to be independent of the empirical world. After all, we can never actually see in nature abstract mathematical entities (such as an infinite flat plane). But, I don’t see this very different feel and style as refuting the claim about the ultimate ontology of mathematics, that it is ultimately an abstracted real-world model in the same way that logic and physics are abstracted real-world models.RobCoel, I can see that the absence of a direct link with empiricism distinguishes mathematics from other areas of science. But you say that you don’t see this

“as refuting [your] claim about “the ultimate ontology of mathematics, that it is ultimately an abstracted real-world model in the same way that logic and physics are abstracted real-world models.”

Well, if that is so, and if it is not about territory, why is there an argument over whether mathematics is part of science.

As an interested lay person I guess I would just like an unequivocal answer as to whether mathematics is science or not. Is the domain of knowledge one or is it not? If it’s ok to have a quid each way, then the argument for scientism would seem less robust.

CoelPost authorSorry, forgot to reply to this. I don’t think one will get an “agreed” answer to this anytime soon! It’s also good to distinguish two issues. The first asserts that mathematics is different from science because it feels different and is done in different university departments (that’s true, but then theoretical cosmology feels different from hunting for fossil whales, and those would also likely be done in different university departments).

The more basic question is the ontology of mathematics, what is it studying? The idea that it is just an arbitrary invention, like chess or Sherlock Holmes, convinces few, since it feels too “real” for that. So, if it is “real”, then it either has to have a real “Platonic” existence distinct from the material world, or it needs to be real by “corresponding” to the material world. I guess the former is the more popular among mathematicians, but it’s a debated question.

On this topic, you may like the recent podcast between (physicist) Sean Carroll and philosopher Jody Azzouni ( link ).

CoelPost authorHi Schlafly, if mathematics and science are distinct, which category would you put Emmy Noether’s theorem in? It was derived by a mathematician in a pure-reasoning mathematical way, and yet it also has profound implications for physics and the physical world. My answer would be that it is “both” mathematics and physics (and thus that the two enterprises cannot be fully distinct).

To claim that it is “not mathematics” because it has real-world applications would place swathes of maths into physics, yet to claim that it is “not physics” because it is mathematics would mean that some of the most profound and important parts of physics are “not physics”. And either of these would leave unexplained why maths works so well as a tool of physics in modelling the world.

SchlaflyNoether’s theorem? Sure, that is math. Just like Newton proving that an inverse square law leads to elliptical orbits.

Noether’s theorem does seem to be part of a broader principle that relates conservation laws to symmetries. That is one of the important principles of XX century physics. Maybe she should have gotten a Nobel prize for that. Except that the prizes are for physics, not math.

RobThanks, Coel. Yes, it’s hard for me to see that mathematics, logic and science are not one. At least, they all seem to be part of the same scientific enterprise. I understand that some philosophers don’t like this view. But I think that may be just philosophical nitpicking and an attempt to demarcate territory. Scientism, the “unity-of-knowledge” view, strikes me as the most accurate and useful way of seeing things.

CoelPost authorMathematicians certainly feel that the edifice of mathematics is “real”, in the sense that it is not just made up, not just an arbitrary invention (in the way that, say, chess is an invention). My explanation for this is, yes, it is real, the edifice of mathematics maps to fundamental truths as to how the external world is at a basic level. The alternative view is to say that, yes, mathematics is “real”, but it is real in a Platonic sense, such that it “exists” as its own Platonic realm that is distinct from the material world that is studied by physics. Thus, mathematical Platonism is a common position among mathematicians and philosophers.

Personally I find this less convincing, because: (1) the idea of a Platonic “existence” distinct from the material world is an idea that does not seem to even have any meaningful content; (2) if there were such a realm, how would we access it, how would we know about it? Our senses gain information from our material world, not from Platonic realms, and the idea that we could intuit the Platonic realm from “pure thought” alone doesn’t seen plausible, and, indeed, since our brains are products of this world, anything our brains intuit must also be intimately connected to this world. Lastly, (3) mathematical Platonism gives no explanation for why mathematics is so useful in modelling the material world. It’s also blatantly obvious that, de facto, over history, mathematicians did not develop mathematics out of “pure thought” alone, they developed it with continual reference to and comparison with real-world behaviour.

Anyhow, I should maybe write another blog post on this if I get round to it. 🙂

RobCoel said:

“I can fully understand why mathematicians feel that maths is distinct from science … It really is very different in style. Mathematicians are using reason (with no direct link to empiricism or data), and arriving at truths that seem to be independent of the empirical world. After all, we can never actually see in nature abstract mathematical entities (such as an infinite flat plane). But, I don’t see this very different feel and style as refuting the claim about the ultimate ontology of mathematics, that it is ultimately an abstracted real-world model in the same way that logic and physics are abstracted real-world models.

That sounds right to me, Coel.

My interest in this debate is not to do with the different styles of mathematics and science but with the deeper ontological question of whether there are separate domains of knowledge that are accessed in different ways and how the answer to this question relates to arguments for and against scientism. If Platonist mathematicians are right, then mathematics is an example of a domain of knowledge that is separate from and not accessed by science and therefore the argument for an entirely scientistic would-view would seem to be weakened. However, whilst mathematicians may not directly use empiricism, the idea that mathematics is done by somehow plugging into a Platonic realm that exists unconnected to the physical world seems to me to be very odd. Like some forms of panpsychism, it smacks a bit of the supernatural for which there is no evidence.

Is Platonism refutable? Can it be tested empirically? If not, then I find it hard to take Platonism seriously. How can a realm that we have no epistemic access to be said to be real? However, if Platonism could be tested, and if it stood up to testing, then scientism might be called into question.

To my mind, what exists is what we have epistemic access to. The property of two-ness, for example, exists and we experience it when we look at a couple of anything, but the abstract concept, the number 2, has no real existence. It’s just a word we invented to refer to the property of two-ness. The written number 2 is a symbol denoting the word we use to refer to the property of two-ness. Numbers are just symbols but the properties they refer to are real. The properties represented by the counting numbers exist in the physical world but the numbers themselves do not. There is no disembodied number 2 floating round in some Platonic realm. Perhaps the axioms of mathematics are like this, too. Much of mathematics has no known applications although applications for pure mathematics are discovered from time to time and this supports the contention that all of mathematics, like the natural numbers, will have roots in real properties of the physical world. I don’t think axioms exist in some Platonic realm. If humans ceased to exist, the property of two-ness would still exist. The number, 2, would not. And, if this is right, then neither would the axioms of mathematics but the properties of the natural world they map onto would continue to exist.

I may not have explained what I mean very well, some of what I have said may be wrong and my reasoning may be shaky. If so, I’d be happy to be corrected and may change my mind. But if what I have said is largely correct, then I think scientism is safe.

CoelPost authorHi Rob, yes, I think that is pretty close to my position.