[Editor’s Note: This essay is part of Scientia Salon’s special “scientism week” and could profitably be read alongside other entries on the same topic on this site, such as this one by John Shook and this one by yours truly. My take on the issue is very different from that of the authors who contributed to this special series, and indeed close to that of Putnam and Popper — as it should be clear from a recent presentation I did at a workshop on scientism I organized. Also, contra the author of the third essay in this series (but, interestingly, not the author of the first two!) I think the notion that mathematics is a part of science is fundamentally indefensible. Then again, part of the point of the SciSal project is to offer a forum for a variety of thoughtful perspectives, not just to serve as an echo chamber for my own opinions…]

by Coel Hellier

While the term “scientism” is often a rebuke to those considered to be overstepping the proper boundaries of science, plenty of scientists will plead guilty to the charge so long as they get a say in how the term is defined. The “scientism” that I defend is the claim that, as far as we can tell, all human knowledge is empirical, deriving from contact with empirical reality. Further, that empirical reality seems to be a unified whole, and thus our knowledge of reality is also unified across different subject areas so that transitions between subjects are seamless.

What we call “science” is the set of methods that we have found, empirically, to be the best for gaining knowledge about the universe, and the same toolkit and the same basic ideas about evidence work in all subject areas. Thus there are no “other ways of knowing,” no demarcation lines across which science cannot tread, no “non-overlapping magisteria.”

A related but different stance is expounded by Pigliucci in his critique of scientism [1]. Pigliucci instead prefers the umbrella term “scientia,” which includes “science, philosophy, mathematics and logic.” This sees mathematics and logic as epistemologically distinct from science. Indeed, Pigliucci has remarked:

“it should be uncontroversial (although it actually isn’t) that the kind of attention to empirical evidence, theory construction, and the relation between the two that characterizes science is ‘distinctive enough’ … to allow us to meaningfully speak of an activity that we call science as sufficiently distinct from … mathematics.”

“… Mathematics is a huge area of knowledge where science has absolutely nothing to say, zip …” [2]

In this piece I argue that mathematics is a part of science. I should clarify that I am taking a broad interpretation of science. Nobody who defends scientism envisages science narrowly, as limited only to what is done in university science departments. Rather, science is conceived broadly as our best attempt to make sense of the empirical evidence we have about the world around us. The “scientific method” is not an axiomatic assumption of science, rather it is itself the product of science, of trying to figure out the world, and is now adopted because it has been found to work.

I will take one statement as standing proxy for the whole of mathematics (and indeed logic). That statement is:

1 + 1 = 2

Do you accept that statement as true? If so (and here I presume that you answered yes), then why?

I argue that we accept that statement as true because it works in the real world. All of our experience of the universe tells us that if you have one apple in a bag and add a second apple then you have two apples in the bag. Not three, not six and a half, not zero, but two. 1 + 1 = 2 is thus a very basic empirical fact about the world [3].

It is a fact about the world in the same way that apples falling downwards are a fact about the world. There is no good reason to place these two different facts (gravity and maths) into two incommensurate domains of knowledge. Our understanding of both derives from empirical reality, and thus both are equally “scientific.”

Having asserted that, let me argue against possible alternative answers to my question of why we accept that 1 + 1 = 2.

*Maths is derived from axioms*

One answer — perhaps the commonest — asserts that maths is not derived from empirical observation but instead derives from axioms. You might assert that you accept 1 + 1 = 2 because it is proven so from the basic axioms of maths. You might point to Peano’s axioms and assert that from those one can logically arrive at 1 + 1 = 2 [4].

My first reply is that I don’t believe you. I don’t believe that there was a time in your life when you were dubious about the assertion 1 + 1 = 2, but then consulted Peano’s axioms, and after some logical thought concluded that, yes, 1 plus 1 really must equal 2. I assert that you accepted 1 + 1 = 2 long before you knew anything about Peano’s axioms, and that you accepted it because it works in the real world: if you had two sweets you could give one to your pal and eat the other yourself [5].

But, even if your belief that 1 + 1 = 2 does derive from axioms, whence your faith on those particular axioms? How and why did Signor Peano arrive at that set of axioms? I assert that they were arrived at with the fact of 1 + 1 equalling 2 being a necessary consequence. Had Peano’s axioms resulted in 1 + 1 equalling anything other than 2 then the axioms would have been rejected as faulty. Signor Peano would have been told to go away and come up with axioms that worked (ones compatible with the non-negotiable truth that 1 + 1 really does equal 2).

Thus, the axioms mathematicians adopt are not arbitrary, chosen by whim or fiat, they are chosen to model the empirical world. Mathematics is thus distilled empiricism. The same can be said about logic and reason. In order to get from Peano’s axioms to derived results you need to use logical reasoning. What validates that logic and that reasoning? Again, I assert that empirical reality validates them. The reason that we adopt logical axioms such as the law of non-contradiction is that they hold in the empirical world [6]. How else would we know which logical axioms to adopt? Thus the whole edifice of mathematics and logic is a distillation of empiricism, created and developed as a model of the basics of how our world works.

*Mathematics is arbitrary*

Nevertheless, some might assert that no, mathematics is a self-contained logical system entirely distinct from empirical reality, and that any correspondence between mathematics and science is simply a coincidence. Some might even assert this with a straight face. It leads to puzzlement over what Eugene Wigner called “the unreasonable effectiveness of mathematics” when applied to science, but there is no puzzle if mathematics describes deep properties of our empirical universe and is derived from that universe. The idea that mathematics is arbitrary and independent of our universe would be more convincing if mathematicians spent as much time pursuing maths based on 1 + 1 equalling six and a half as they do with 1 + 1 = 2.

A more sophisticated version of this answer accepts that mathematics originally derived from our universe (with, for example, Pythagoras’ theorem resulting from drawing on bits of paper, or from attempts to get a building’s walls square), but points out that nowadays mathematicians experiment with all sorts of axioms that are not first suggested by observation.

As an example, consider the generalization of the “flat” geometry developed by Euclid to the “curved” geometries developed by Carl Gauss, Bernhard Riemann and others. The relaxing of the parallel-line postulate of Euclid to produce non-Euclidean geometries was not motivated by observations but by thinking about the structure of the axiomatic system. Surely this is a non-empirical approach that distinguishes mathematics from science?

Well no. Theoretical physicists do this sort of thing just as much as mathematicians. They take their set of empirically derived axioms (though in physics these tend to be called “laws” rather than “axioms”) and think about them; they experiment with different axioms/laws and work out the consequences. Often they are not immediately motivated by a match to observations but are following their intuition.

They are still, though, working with an axiomatic system that is essentially distilled from the empirical universe, and they are using an intuition that is also very much a product of the empirical universe. Curved geometry — developed by the mathematician Riemann — was later found to be useful in describing the universe when the physicist Einstein — also following a path of logic and intuition — developed the theory of General Relativity. If anyone wants to draw a demarcation line between domains of knowledge, the line would not be between the mathematician Riemann and the physicist Einstein.

Why is it that mathematicians’ intuitions so often produce mathematics that is later found to be useful to physicists? I argue that their experimentations with axioms are productive because their logic and intuitions are also empirical products. Thus a mathematician has a good idea of which changes to axioms are sensible and which are not. Allowing parallel lines to diverge (and thus producing non-Euclidean geometry) is sensible; adopting “one plus one equals six and a half” is not [7]. In both mathematics and physics, if the experimentation produces results that are nonsensical when compared to our universe then they will not be pursued. The empirical universe is in both cases the ultimate arbiter.

At the cutting edge it can, of course, be unclear whether maths and/or physical theories “work.” A current example is string theory, where a generation of theorists is exploring the mathematics of strings. Maybe it’ll lead to new physical theories unifying quantum mechanics and gravity, and maybe not. At the moment, though, one could not really say whether string theory was “mathematics” or “theoretical physics.” This emphasizes the seamless transition between those fields, with string theory straddling the (arbitrary and unmarked) boundary.

An aside before proceeding. Gödel’s incompleteness theorem tells us that even if we have a set of axioms such as Peano’s, which underpin the natural counting numbers and which yield the statement that 1 + 1 = 2, there will be other statements about the natural numbers which are true, but which cannot be shown to be true from the axioms. A further result tells us that the axioms cannot be used to show that the system built from those axioms is consistent. This fundamental limitation of an axiom-based approach shattered hopes of mathematics ever being a complete, consistent, self-validating and self-contained system.

From a scientific point of view, with mathematics being seen as a part of science, such limitations are unsurprising. Science is derived from empirical evidence and our available evidence will always be a small and incomplete sample of the universe, and thus scientific results are always provisional, in principle open to revision given better data.

*Our math is the product of pure logic, deriving only from human intuition*

Many will disagree with me and assert that human intuition is a primary source of knowledge, distinct and separate from empirical evidence. Indeed this idea is popular with some philosophers, who argue that introspection and thought are the wellsprings of their philosophical knowledge, and thus that philosophy is a domain distinct from the empirical domain of science [8].

However, what basis do we have for supposing that our human intuition produces accurate knowledge about the universe? The first reason is that our intuition has been developed and honed over our lives based on our sense data about the world around us. Thus our intuition is very much an empirical product.

Further, we can ask about instinct, that portion of our intuition that is not the product of life experiences, but is encoded in the genes. Our genetic programming will also be a product of empirical reality. Our brains are the product of evolutionary natural selection, and thus have developed to make real-time decisions that aid survival and reproduction. Obviously, decision-making that bore no relation to the real world would be useless, and thus we can have some confidence that our intuitions are to a large extent programmed to produce decisions well-aligned to empirical reality.

Of course natural selection is not a perfect programmer, and anyhow is not aiming at a perfect and unbiased decision-maker, it is aiming at the one best at survival and reproduction. Thus we would expect our intuition to be reliable only with respect to the everyday world relevant to survival and reproduction, and to be unreliable about aspects of the universe (such as quantum mechanics and general relativity), that are irrelevant for everyday life.

We’d thus expect our intuition to be a folk metaphysics, good enough for many purposes, but full of biases and foibles, particularly so where an inaccurate assessment might actually aid survival and reproduction. An over-active pattern-recognition detector and the Lake Wobegon effect are likely examples of this. Visual illusions such as the checker-shadow illusion [9] show how easily human intuition is fooled, in this case precisely because it is making some assumptions about how the world works, and thus about lighting and shading.

A critic might, though, accept that some of our intuitions are related to empirical reality, but argue that intuition also gives access to knowledge that is not empirical and cannot be arrived at by empirical means. My response is to ask what basis the critic has for that assertion and what reason he has for supposing that “non-empirical knowledge” has any reliability or validity.

From the evolutionary perspective we have no good reason to suppose that intuition is anything other than an imperfectly and empirically programmed device that models the empirical world — after all, failing to find enough to eat, ending up eaten by a predator, or finding a mate and successfully rearing children, are all aspects of a brute empirical world. Thus we should accept intuition as a useful “quick guide to reality,” but ultimately we should not accept it except where corroborated by empirical evidence. Indeed, the whole point of the scientific method is to use empirical evidence to do much better than just consulting our “quick guide” intuition [10].

*Our math is the only possibility*

The last alternative answer that a critic might advance is that we accept the claim that 1 + 1 = 2 because it must be true, it is the only logical possibility. Thus, such a critic will say, 1 + 1 equalling six and a half is simply nonsensical. Such a person would not merely be asserting that it is impossible in our world, but that it is impossible in all possible alternative worlds.

Do we know this? And, if so, how? Has anyone given a logical proof of the impossibility of such an alternative scheme? Any such proof could not use any axiom or logic derived from or validated by *our* empirical world (that would only show that such alternatives did not occur within our world). But without that, how would one go about showing that the logic of our world is the only one possible?

One could not use our-world logic for such a task, nor could we use human intuition, since our intuition is very much derived from and steeped in the logic of our own empirical world — indeed our brains have evolved precisely to model the logic of our world — and thus we would not expect them to be in any way useful for contemplating radically different alternatives.

But, even if we were to grant the claim that our world’s logic is the only possible system of logic, that would still leave the question of how *we* came to learn about that logic. And the only plausible answer is that we learned from observation of the empirical universe and thence deduction about the logic by which it operates.

*Conclusion*

I have argued that all human knowledge is empirical and that there are no “other ways of knowing.” Further, our knowledge is a unified and seamless sphere, reflecting (as best we can discern) the unified and seamless nature of reality. I am not, however, asserting that there are no differences at all between different subject areas. Different subjects have their own styles, in a pragmatic response to what is appropriate and practicable in different areas. For example, a lab-based experimental science like chemistry has a very different style to an observational science like astronomy [11]. Further, biochemists studying detailed molecular pathways in a cell will have a very different style from primatologists studying social interactions in a wild chimpanzee troop.

Such differences in style, however, do not mandate that one of those subjects be included within “science” and another excluded. The transitions in style will be gradual and seamless as one moves from one subject area to another, and fundamentally the same basic rules of evidence apply throughout. From that perspective mathematics is a branch of science, in the same way that so is theoretical physics. Indeed, some theoretical physics is closely akin to pure maths, and certainly far closer to it in style and content than to, say, biochemistry. The different subject labels can be useful, but there are no dividing lines marking the borders. No biochemist worries about whether she is doing biology or chemistry, and string theorists don’t worry much whether they are doing maths or physics.

Thus, in arguing that a subject lies within the broad-encompass of “science,” one is not asserting that it is identical in style to some branch of the generally-accepted natural sciences, but that it belongs to a broad grouping that spans from studying molecules in a chemistry laboratory, to studying the social hierarchies of a baboon troop, to theoretical modeling of the origin of the universe, and that it belongs in that group because epistemologically the resulting knowledge has the same empirical source.

I thus see no good reason for the claim that mathematics is a fundamentally different domain to science, with a clear epistemological demarcation between them. This same set of arguments applies to the fields of reason and logic, and indeed anything based on human intuition. All of these seem to me to belong with science, and all derive from our empirical experience of the universe and our attempts to make sense of it.

_____

Coel Hellier is a Professor of Astrophysics at Keele University in the UK. In addition to teaching physics, astrophysics, and maths he searches for exoplanets. He currently runs the WASP-South transit search, finding planets by looking for small dips in the light of stars caused when a planet transits in front of the star. Earlier in his research career Coel studied binary stars that were exchanging material, leading up to his book about Cataclysmic Variable Stars.

[1] Massimo Pigliucci, Midwest Studies in Philosophy 37 (1):142-153 (2013) “New Atheism and the Scientistic Turn in the Atheism Movement.”

[2] See Pigliucci’s article, Staking positions amongst the varieties of scientism.

[3] A pedant might point out that in modular arithmetic, modulo 2, 1 plus 1 would equal 0. I am taking 1 + 1 = 2 to refer to simple counting numbers; one apple plus one apple equals two apples. If we ask further about the basic concepts of “1,” “2,” “+” and “=” I would again base them on patterns discerned in the empirical world, which is of course how humans first came up with those concepts.

[4] Giuseppe Peano, 1889. Arithmetices principia, nova methodo exposita*.*

[5] A pedant might point out that that equates to 2 – 1 = 1, not to 1 + 1 = 2.

[6] Indeed the great Islamic polymath Avicenna wrote, circa AD 1000, that: “Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned,” a direct derivation of logic from empirical experience!

[7] The Axiom of Choice is an example of an axiom adopted largely because it feels intuitively right to mathematicians, plus they like the results that it leads to.

[8] There is a vast philosophical literature on this issue, with Kant’s Critique of Pure Reason being influential.

[9] For the Lake Wobegon see here. For the checker shadow illusion see here.

[10] An obvious example being the need for double-blinding in medical trials, which originated from the realization of how unreliable human intuition, based on anecdotes and a partial memory, actually is.

[11] One should ignore commentators who over-interpret overly-simplistic accounts of the “scientific method” and claim that only lab-based experimental science counts as science.

Perhaps this.

There isn’t a scientific instrument that is going to give you a readout with an actually infinite number of decimals, but can π be what is being measured here?

“Pi helps describe the shape of the universe, says David Spergel, chairman of Princeton University’s astrophysical sciences department.”

“4π is the ratio of the surface area of a sphere to the square of its radius, in geometrically flat space.”

“Using our measurements of the microwave background, we measure by determining the angular size of hot and cold spots in the microwave sky. Our measurements show that the large-scale geometry of the universe is accurately described by the Euclidean geometry that we all learned in high school,” Spergel says. “This measurement implies that the total energy of the universe is very close to zero.”

http://lightyears.blogs.cnn.com/2012/03/13/pi-day-how-3-14-helps-find-other-planets-and-more/

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Your admission that “maths nowadays, is just the reasoning, and is not a matter of doing observations” ought to settle the matter, because we are talking about math as it is practiced today, not as it originated. If you go far back in time, there was no math, no science, no systematic human quest for understanding. Of course this makes a (huge) difference epistemologically.

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Coel, you would have to go back millennia to find that. Archimedes found that pi was between 223/71 and 22/7 in about 250BC, and he did it using math, and not experiment or observation.

Apparently your arguments reduces to saying that ancient pre-Greek math was based on observation and subject to empirical test. The Greeks made a wonderful invention of the mathematical proof. You are in denial that this ever happened, as you insist that all math is provisional, just like the inverse square law of gravity. It is not. Math proofs achieve a certainty that is not possible in the study of exoplanets, or whatever your field is.

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Agree with you Schlafly, thought I would replace “invention” with “discovery”

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stevenjohnson,

Your reply indicates that we have more in agreement than opposed.

I will only note that I have no interest in waging war against science. Let all sciences pursue their investigations. I for one am learning to learn from them. My suspicion, however, is that individual experience will elude them all. But certain social discourses, e.g., aesthetics, necessarily include individual experience, and they’re all we have concerning such or now. As for the future, we shall see.

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Corrections to previous post:

I am *willing* to learn from the sciences.

(Certain social discourses) are all we have *for* now,

Sorry for the mistakes, it’s late.

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Hi Massimo,

Hardly an “admission”, it’s entirely obvious that pure mathematicians spend their lives by reasoning from axioms (as opposed to making comparisons to the real world). So can many theoretical physicists. On the other hand, applied mathematicians are continually making real-world comparisons. All of these are differences in style, but are not what my thesis is about.

My argument is about epistemology, about how we obtain knowledge. For that the different styles and the timing are not the issue. For both maths and physics we develop models, encoded in axioms & laws, and those models/axioms/laws are arrived at empirically. Everything in maths and physics thus uses a mixture of observation and reason, and differences in the proportions or the timings are just differences in style.

Thus the argument is about sources of knowledge, and about the different domains of knowledge being unified and seamless — since they derive from the same source — and is a rejection of different domains being epistemologically independent, deriving from “other ways of knowing”.

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Hi schlafly,

You would indeed. So what? Why does three thousand years versus ten years make such a big difference to the epistemological source of the knowledge?

Correct, and the maths that he used derived from observation.

Not at all. Physics uses proofs all the time also.

Maths is derived from real-world observation and thus is technically “provisional” in the same way that physics is. But, some things are sufficiently well established that it really is unlikely that they’ll ever be over-turned. The scientific statement that “the earth’s shape is nearer that of a sphere than that of a flat pancake” is technically provisional, but I doubt if it will ever be over-turned.

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“Archimedes found that pi was between 223/71 and 22/7 in about 250BC, and he did it using math, and not experiment or observation.”

Though if he wanted to know what the 12,124,782,995,784th digit of pi was, he have to build a big and fast enough computer, run a program, and then observe the result.

http://www.numberworld.org/misc_runs/pi-12t/

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Hi Philip,

I think that you have a very good point about mathematics done by computation (which is an increasing feature of maths today). Massimo wants to distinguish the “concept” of pi from the “physical device that calculates it”. But why does the computational calculation of it actually work?

I suggest that it can only work if the conceptual maths is a real-world model, and thus uses the same basic logic as the real-world compational device. In other words there is a seamless meshing between the real-world behaviour and the maths, and that allows one to obtain digits of pi either by computation, or by playing with polygons internal to a circle, or by getting a piece of string and wrapping it around the circumference of a circle.

All of this meshes seamlessly, and that can only be the case if the same epistemology consistently underlies everything.

(In contrast, one could not, for example, demonstrate Banach-Tarski on a computational device.)

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Epistemology is about how propositions are justified. The study of how they are acquired is psychology.

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>>I suggest that it can only work if the conceptual maths is a real-world model, and thus uses the same basic logic as the real-world compational device. In other words there is a seamless meshing between the real-world behaviour and the maths, and that allows one to obtain digits of pi either by computation, or by playing with polygons internal to a circle, or by getting a piece of string and wrapping it around the circumference of a circle.

But that’s not the case. The math doesn’t serve as a real-world model, since space isn’t Euclidean. If you measured the circumference/diameter ratio of a disc the size of a galaxy, for example, you wouldn’t get the same value as those computations of pi. This doesn’t mean there’s anything wrong with the methods for computing pi, though, because the value of pi has nothing to do with the structure of physical space.

>>(In contrast, one could not, for example, demonstrate Banach-Tarski on a computational device.)

Yes it can. It’s called automated theorem proving.

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I thought about writing this very same article. I think Coel is absolutely right on this subject and it’s interesting to me how everyone else on the forum seems to have so much trouble with it. Thanks Coel for articulating the position so clearly. I hope other readers can appreciate the exposure to the alternative conception of mathematics and science that I think is much more coherent and much more fecund for research in the future.

I haven’t much else to say in support of Coel’s position. This statement is just for moral support. Some of us are listening, Coel! Even if it’s only the younger one’s of us less deeply entrenched in the traditional philosophy 😉

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Hi Coel,

Nicely written, and I totally agree.

To Massimo: Was Einstein doing science or math when he was sitting on the train? I think he was making predictions based on previous observations of the world. Change those previous observations and you have to change the predictions. Coel is saying (I think) that today’s mathematicians are making predictions based on previous observations (way, way, previous) of the world, and changing those observations, I.e., changing the physics, will change the predictions, I.e., change the math.

In support of Coel’s position , I offer David Deutsch’s Constructor Theory of Information (Google it) which proposes that information, often thought of as mathematical in nature, is actually necessarily dependent on the laws of physics.

James

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Coel’s whole argument hinges on the fact that 3000 years ago some Babylonian wrapped a string around a circle, and approximated pi as 3. He denies that mathematicians have proved anything since, as he claims that all math results are just provisional, like an inverse square law. He is about 2300 years out of date.

He is like a mathematician posting a rant against the reliability of cosmology or the scientific method, and then basing the whole argument on criticisms of Aristotle, and arguing that everything since is derivative.

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I guess you are going to say that Einstein’s famous 1905 relativity paper was doing math, not science, because he reformulated the Lorentz theory as being based on postulates, instead of on the Michelson-Morley experiment. Einstein did not do any physical experiments, or predict any observations beyond what Lorentz already predicted. However he did not state or prove any mathematically precise theorems, and that is usually what is meant by math research.

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Hi James and Chance, thanks for your support and comments!

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“Coel’s whole argument hinges on the fact that 3000 years ago some Babylonian wrapped a string around a circle, and approximated pi as 3.”

This is a straw man of his position, whether you intend it to be or not. His claim has only contingently to do with the history of mathematics (if at all). His real claim is that the axioms of mathematics are verified by the senses, which means all of mathematics could be derived independently of our provincial history. Were this not the case, tropes about mathematics being “the language of the universe,” would be senseless, and SETI wouldn’t consider prime numbers to be a likely signal from extra terrestrials.

“He denies that mathematicians have proved anything since,…”

I assume you’re speaking to the implications of his article and not any explicit statements he made. If the former, I disagree on the grounds I’ve tried to communicate above, and if the latter, then please provide a direct quote of him stating that no mathematicians have proved anything since.

“…he claims that all math results are just provisional, like an inverse square law. He is about 2300 years out of date.”

On some level every human constructed law and belief must be provisional. If a claim is not provisional, then it cannot claim to be representing reality. Claims about reality must be a *function* of the reality and not vice versa, thus all claims must be provisional or they can no longer claim to be *about* reality.

“He is like a mathematician posting a rant against the reliability of cosmology or the scientific method, and then basing the whole argument on criticisms of Aristotle, and arguing that everything since is derivative.”

On the contrary, I see your argument like one which insists on talking about color as an abstract folk concept that “just is” while refusing to accept any discussion about how color is actually a product of the wavelengths of light.

Again, Coel’s claim has nothing to do with our provincial history of mathematics. It’s a simple statement verifiable today independently by anyone who tries to create a formal system of logic/number theory: All mathematics is derived from experience, just as all of any human construct is derived from experience. Mathematical truths aren’t popping into our minds out of thin air, and they’re not “discovered” independently from those very minds either, so what is the basis of mathematics? Just like any other human product, it first arose in physical reality where it could be experienced, processed, and finally conceptually manipulated by our minds to produce other axioms and systems of logic that needn’t have any independent basis in external reality themselves. I hope that gains some traction with you.

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This (and Coel’s entire position) confuses the genealogy of a belief with its justification, a fundamental distinction in epistemology.

The fact is that I acquired my belief that 1+1=2 from the utterances of my kindergarten teacher. But those utterances are not what justifies the proposition that 1+1=2.

Now, there *are* people who want to naturalize epistemology, which essentially involves rejecting the distinction between the genealogy and justification of a belief — and thus, is tantamount to rejecting the concept of justification itself — but this is a *highly* controversial position, on which there are a lot of arguments that would have to be worked through. It is hardly something that someone can just assert.

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Coel,

You might be interested in what Terence Tao has to say about the Banach-Tarski paradox (expressed in the computer language of oracles):

https://terrytao.wordpress.com/2010/03/19/a-computational-perspective-on-set-theory/

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Aravis, I want to emphasize again that this claim (as I would support it) has nothing to do with genealogy and everything to do with, as you suggest, *how* claims get justified. Systems of logic and mathematics are not self justifying. However remote mathematics/logics go beyond exactly *how* they are justified is through empirical observations. They may then go on to prove truths using their own internal logics, but without first demonstrating the value of the number, addition, subtraction, division, and all other patterns *observable* in human categorization and understanding, they’re not going to have value in the first place. It would be incoherent babble if you invented a “formal” system of random numbers and characters and had no laws behind them which were first rooted in the empirical world.

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However remote mathematics/logics go beyond what we can actually observe is irrelevant, exactly *how* they are justified in the first place is through empirical observations.

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I’m sorry, but having taught logic now, for some 20 years, your description of the subject is not something I find recognizable.

There is nothing “incoherent” or “babble” about a purely formal system, in which values are assigned to variables. Indeed, I can create indefinitely many such formal languages, none of which need be incoherent or babble.

And yes, it seems pretty clear that you (and Cole) *are* talking about genealogy–or better, etiology–and not justification, as the concept is used in Epistemology. I’m sorry you don’t like it, but there it is. And this is likely the reason why you will get so little agreement from those whose professional training is in Epistemology and from teachers of logic. These distinctions are simply basic to what we do. And it is worth noting that even those who wish to naturalize epistemology and logic, like Quine, realize that in doing so, they have to sacrifice the concept of justification. The trouble is, when you drop the concept of justification, it becomes difficult to make sense of the normativity of reason and ultimately, of truth.

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Evidence of the conflation of etiology of belief and justification, from the article itself:

——-

“One answer — perhaps the commonest — asserts that maths is not derived from empirical observation but instead derives from axioms. You might assert that you accept 1 + 1 = 2 because it is proven so from the basic axioms of maths. You might point to Peano’s axioms and assert that from those one can logically arrive at 1 + 1 = 2 [4].

I don’t believe that there was a time in your life when you were dubious about the assertion 1 + 1 = 2, but then consulted Peano’s axioms, and after some logical thought concluded that, yes, 1 plus 1 really must equal 2. I assert that you accepted 1 + 1 = 2 long before you knew anything about Peano’s axioms, and that you accepted it because it works in the real world: if you had two sweets you could give one to your pal and eat the other yourself.”

——–

Pretty transparent. And nowhere in the article is there even an inkling that the author understands the distinction.

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Just to clarify, “However remote mathematics…” was meant to replace the sentence of the same beginning in the previous post but I screwed up. It should read a lot better now if read that way.

“There is nothing ‘incoherent’ or ‘babble’ about a purely formal system, in which values are assigned to variables.”

Correct, and I would not dispute this. Your expertise is helping us clarify things here. What I said was not exactly as you rephrased it. What I said was this:

“It would be incoherent babble if you invented a ‘formal’ system of random numbers and characters and had no laws behind them which were first rooted in the empirical world.”

In your phrasing, you’re again depending on rules and laws of language rooted in empirical experience to formulate the axioms, laws, and postulates of your new system. Again, mathematics and logic are at their root fundamentally empirical.

We (Coel and I, I presume) are talking about the epistemology of the basis of mathematical and logical systems. Proofs are then etiologically derived from such systems, so on some level we are talking about etiology, yes, but the primary thrust is that all mathematics is derived out of the human mind which is constructed and reshaped through long evolutionary and short lifetime empiricism. There is just no escape from physical reality. It must and does provide the basis for all understanding, whether it be “scientific” or “mathematical.”

“The trouble is, when you drop the concept of justification, it becomes difficult to make sense of the normativity of reason and ultimately, of truth.”

I see no need to abandon justification in Coel’s “scientism” or in my apparently analogous conception of the relationship between science and mathematics. In fact science has justification on lock.

“Evidence of the conflation of etiology of belief and justification, from the article itself: […] Pretty transparent. And nowhere in the article is there even an inkling that the author understands the distinction.”

I guess I may be giving the author too much credit, but that is not how I read it. At no point does he say that because his beliefs in 1+1=2 (for example) were caused by empirical observation they were then justified by empirical observation, though I agree it appears to imply so. Instead, when I read that passage I considered 1+1=2 to be a stand in for the generalized axioms of mathematics. This to me was the thrust of his argument, and he was simply using 1+1=2 as an example a mathematical truth that _does in fact_ have an empirically observable basis. In other words, he doesn’t *have* to be confusing anything here, it could just be reading comprehension on your end.

On the other hand, and to your credit, does he mention anywhere in the article that he understands the distinction as I’m attempting to express it and as you’re attempting to point out?

Let me look… Well first off he does say this:

“I will take one statement as standing proxy for the whole of mathematics (and indeed logic). That statement is: 1 + 1 = 2.”

This strongly implies my interpretation. Secondly the section titled, “Our math is the product of pure logic, deriving only from human intuition” is to some degree an attempt at debunking the idea that etiology of belief is a justification for it. For example in this section he says,

“Of course natural selection is not a perfect programmer, and anyhow is not aiming at a perfect and unbiased decision-maker, it is aiming at the one best at survival and reproduction. Thus we would expect our intuition to be reliable only with respect to the everyday world relevant to survival and reproduction, and to be unreliable about aspects of the universe (such as quantum mechanics and general relativity), that are irrelevant for everyday life.”

Here it’s stated that our empirical intuitions aren’t enough to justify beliefs.

And further, “Thus we should accept intuition as a useful ‘quick guide to reality,’ but ultimately we should not accept it except where corroborated by empirical evidence. Indeed, the whole point of the scientific method is to use empirical evidence to do much better than just consulting our ‘quick guide’ intuition [10].”

Though I did not see any explicit attack on the distinction between etiology of belief and the justification of belief, it is heavily implied by the (scientifically informed) logic of his argument in this section.

Either way, we don’t need to consult his article to come to the conclusion as I stated above: the systems of axioms and laws of mathematics and logics are empirically formulated and then their consequences are etiologically derived and stated as the proofs and postulates we all know and love.

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Hi Aravis,

The quote you give does not support your claim. I explicitly say that someone regards the model 1+1=2 as *justified* because it works in the real world: “I assert that you accepted 1 + 1 = 2 long before you knew anything about Peano’s axioms, and that you accepted it because it works in the real world: if you had two sweets you could give one to your pal and eat the other yourself.”

That is not saying that you “know” it because your kindergarten teacher taught it to you, nor that we know it because it follows from Peano’s axioms, it is saying that we *justify* it “because it works in the real world”.

That whole essay *is* about justification.

And schafly is right that the relevant observations were first done eons ago, but they have been repeated in every generation since. Each generation uses maths because it works, each generation uses technology based on maths, and each generation would start noticing if maths wasn’t a good model of real-world behaviour. *That* is the justification for maths.

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Hi Coel,

This discussion is way past its prime … but we’re approaching things too much from the mathematical side. It’s the physical side that’s interesting.

Say you move in the X-direction with a speed of (0.9)c, c being the speed of light. I move in the –X-direction with a speed of (0.8)c. Therefore, our relative speed is (1,7)c. I could also use special relativity and arrive at another result.

Could you give me a mathematical reason why I should use special relativity?

And if you can’t give it, where does that leave you claim that mathematics and science are epistemologically equivalent?

In both cases (classical mechanics and special relativity) the mathematics is impeccable. Galilei and Einstein start from a different symmetry group, but on a fundamental, axiomatic level they’re using the same mathematics: the natural numbers with addition and some naïve set theory. Everything Einstein and Galilei needed mathematically can be derived from that starting point.

So how come that scientifically speaking Galilei is wrong and Einstein correct? Does mathematics tell you which symmetry group to use? Answer: no, it doesn’t. It’s nature and not mathematics that indicates which symmetry group is the correct one. Mathematically speaking, Einstein and Galilei are just as correct – they use exactly the same axioms and exactly the same logical rules.

Mathematics and science cannot have the same epistemology.

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Or let me put it this way …

If a mathematician says “let’s assume X is true”, what s/he really says, is: “Y follows from X” is mathematically correct.

If a scientist says “let’s assume X is true”, what s/he really says is: “let’s assume that X is a correct scientific description of the real world.”

Not epistemologically equivalent, if you ask me. Not in a million miles.

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What Patrick says…

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Hi Patrick,

“Relative speed” is not a pure-maths concept, it is a physical concept. Thus you need both maths and physics (= applied maths).

I agree entirely. Some aspects of nature are best modelled by some parts of maths, and other aspects of nature are best modelled by other parts of maths.

I don’t see how that follows from anything that you’ve said. All that you’ve said is that you can write down a mathematical model that does not work well in a particular physical situation. So what? You can also write down physical models that do not work well in particular physical situations.

For example, Newtonian gravity works well in some situations but not in others, just like the Galilean transforms that you point to work well in some situations and not in others, and in some situations the Lorentz transforms work better.

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Coel, Patrick’s point is that “works better” makes sense only when you apply mathematical models to physical problems. But the very notion is nonsensical when it comes to the math considered on its own. At that level, the only question one can ask is whether the models are coherent. That’s why science and math are epistemologically different: roughly speaking, in the first case one applies a correspondence theory of truth, in the second a coherence theory.

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Hi Patrick,

Agreed.

But you’re not addressing the full picture! The physicist *also* uses reasoning and logical deduction.

For example, the physicist can reason: Let’s assume that the postulates of relativity are true. From that, purely by reasoning, we can arrive at “Lorentz contraction and time dilation occur”. One can then ask whether those phenomena are a good match for real-world observed behaviour.

Thus there are two aspects to the “is it correct?” question. These are:

1) Does this result follow from the postulates/axioms?

2) Does the result hold in the real world?

Now, if (1) holds then (2) is equivalent to (2a) “Do the postulates/axioms hold in the real world”?

The physicist is asking both of those questions.

Now, you are entirely right that the mathematician can just concern themselves with (1). And that point, that mathematicians spend most of their time only doing (1) is blatantly obvious.

But, that leaves the axioms unexplained. And anyone with any curiosity then asks themselves where the axioms came from, why mathematicians work with *those* axioms. That is surely fundamental to the whole of maths, and fundamental to the epistemology of maths.

Some here seem to be of the opinion that if they simply don’t ask that question, regarding the axioms as a given that they don’t question, then by doing so they divorce maths from the empirical world. Which seems to me very incurious — they fail to see the link to the empirical world purely because they refuse to ask that question!

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Coel,

Aravis explained to you very clearly the difference between the etiology and justification. You really should pay more serious attention to the arguments of skilled, professional philosophers.

But let’s put that aside for the moment.

I have already explained to you the cryingly obvious, that mathematical knowledge is scaffolded, rising far above its slender empirical foundations. So far your arguments have only dealt with these foundations.

Now my question to you is this. On the upper stories of the scaffold that represents mathematics, how, or in what way does that resemble science? This is the real question and you have not addressed it at all. If you fail to address that your arguments are null and void.

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Dear All,

I’m guessing that before long the curtain will come down on this thread. Thanks to Massimo for scientism week, it was fun, and thanks especially to Massimo given that he totally disagrees with scientism!

Here is a little story to sum things up:

A scientist sits by a window observing the world. She sees Jack approach, and declares: “I see that Jack is carrying an apple”. On looking further she declares: “I also see that Jack is carrying an orange”.

A mathematician, sitting inside and not looking out, overhears. The mathematician then declares: “We can deduce that Jack is carrying at least two pieces of fruit”.

A philosopher is also sitting inside observing the physicist and the mathematician. The philosopher declares that the mathematician has used only reasoning; the mathematician has not looked out the window and not made empirical observations, but has only reasoned logically from statements given.

Thus the philosopher declares that the mathematician’s knowledge has a fundamentally different source, and that the mathematician’s knowledge has nothing at all to do with the empirical world!

On hearing that, the scientist, rather baffled by this, points out that the statements that the mathematician used were derived from the real world, and thus that the statement “Jack is carrying at least two pieces of fruit” is ultimately derived from, and thus still eminently about, the empirical world.

The philosopher then rather grudginly admits that the statements from which the mathematician started did indeed have some tangential and historical connection with the natural world, but then declares that to be irrelevant. That is simply not the mathematician’s concern. The point — the philosopher insists — is that the *mathematician* has not directly observed Jack, and thus anything the mathematician says has nothing at all to do with the empirical world!

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Hi Massimo,

The reply is that science *also* uses theorizing, and thus science *also* concerns itself with the coherence of the models.

Thus science includes *both* the mathematical-truth aspect of truth and the correspondence-with-reality aspect of truth.

One could regard maths as a subset of the overall scientific enterprise, being concerned with reasoning internal to axiomatic models, but the fact that those models derive from the real world and were adopted because they are good real-world models makes maths part of the overall enterprise of modelling the real world that we call “science”.

In that sense, note my title that “mathematics is a *part* *of* science”. Maths can still be a part of science even though the mathematicians themselves don’t concern themselves with the correspondence to the real world.

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And that’s the crux of our disagreement: *of course* science uses coherence, but it is a coherence that is constrained by empirical information. Math’s isn’t. It really is that simple.

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Hi labnut,

Sure, and he is simply wrong in suggesting that my article was not about justification. All of it was about justification

It sure is!

No, I’ve explicitly replied to your point. If the foundations are empirical, and if the edifice is tautologically entailed by the empirical foundations, then the whole edifice is still about the empirical world.

That seems to me “cryingly obvious” (to use your expression).

It is still a real-world model! Science *also* uses reasoning from empirically derived axioms. If the edifice of reasoning follows logically from the real world then the edifice is still a real-world model, still about the real world.

It seems to me that I have explicitly answered your question about 15 times!

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If the axioms of maths are derived from (and thus constrained by) the empirical world, and if the edifice of maths follows tautologically from those, then maths surely *is* constrained by empirical information. Maths is tied to empiricism by tautological tethers!

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And here we go again in circles! *Some* axioms of math may have historically been arrived at empirically. But they do not need any empirical support in order to be coherent. And most mathematical results are not at all meaningfully connected to empirical results. And the same goes for logic, about which you have remained strangely silent throughout.

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Which maths axioms are not arrived at owing to their correspondence with the real world?

So far no-one has pointed to a single axiom of maths that has no real-world correspondence.

Now, if lots of maths axioms are indeed adopted that are totally counter to real-world behaviour, then I agree that it refutes my thesis. But which axioms are those?

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Coel, every time someone has suggested an answer to that question you have simply made up an empirically unsubstantiated story about how people may have derived axiom X or Y from the real world. It has been pointed out to you repeatedly that plenty of mathematical structures simply do not describe the world, some miss by little (Euclidean geometry, when used locally), some have nothing whatsoever to do with it (multidimensional topologies. I am really having a hard time understanding why you insist in denying the obvious. Which, of course, it is your prerogative to do.

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Hi Massimo,

Sure, agreed, from mathematical axioms one can build many mathematical structures that are not instantiated in the world. And from physical laws one can build many physical structures that are not instantiated in the world.

Those both follow from the fact that maths axioms and physical laws are deep regularities in nature, and only a subset of possible things allowed by those laws/axioms is actually instantiated.

But mathematical *structures* are not the issue here, we’re arguing about the basic axioms from which the edifice is built.

Which mathematical axioms have no correspondence to real-world behaviour? My assertion is that mathematical axioms are adopted as distillations of real-world deep regularities.

Perhaps one or two you can argue about (e.g. axiom of choice applied to infinities), but so far I’m sticking to the claim.

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Massimo, I’m with you on this but I don’t think Coel’s position is that hard to understand. Coel’s view is that abstract mathematical concepts such as multidimensional topologies are extrapolated from observation of the physical world. I don’t agree with calling them empirical, but while I disagree with him his position seems at least arguable and not all that bizarre.

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Hi Coel,

The analogy is flawed. The example you gave was applying mathematics to the real world, which of course has rather straightforward empirical connections.

The mathematician should not be saying “Jack is carrying at least two pieces of fruit.”. The mathematician should be saying that wherever there exist an object and another object, there are at least two objects.

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Parallel postulates. Ever seen straight lines that go on infinitely and maintain parallelism? Where?

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DM, I don’t think his position is hard to understand, I just think he helps himself conveniently to approximations and ignores flagrant counter-examples.

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That’s a notable example, since of course it only holds in some situations, and for large swathes of maths one does not adopt that axiom.

But, yes, it does have a real-world correspondence. If you draw two parallel lines on a flat sheet of paper that is what happens. And I’m willing to bet that Euclid wrote down that axiom from considering such real-world behaviour. (What do you think led to Euclid adopting that axiom?)

Of course the real-world behaviour does not extend to infinity, so the axiom is an abstraction of that. But then so are all laws of physics!

Have you ever seen an inverse-square law? That extends to infinity also, but no-one has seen it extending to infinity, the law is an abstraction of real-world behaviour.

So, your example of the parallel postulate is a very good example of an axiom that is indeed adopted from consideration of real-world behaviour and adopted precisely because if its real-world correspondence.

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Hi DM,

But the mathematician did not try to “apply” his reasoning to the world. He did not look out of the window to verify the statement.

All that the mathematician did was reason from statements adopted as axioms. And that, I am told, is an activity that has nothing to do with the empirical world!

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I agree that mathematicians are doing empirical science: they are discovering the objective world of mathematical structures. There is obviously some overlap between this mathematical world and our physical world (it may even be that our physical world is just a subset of the mathematical world, as is suggested by Max Tegmark), so sometimes mathematicians discover facts about our physical world too, which can be supported by physical evidence. But in any case they discover facts about objective reality.

The mathematical world seems to be a necessary objective reality. It is described by set theory, which is based on the self-evident assumption that any objects necessarily form a set (a collection), with the exception of cases where this would lead to a contradiction (e.g. Russell’s Paradox). The special case of a set is the empty set, which is formed by no objects (that is, by nothing). A set may also be interpreted as a fact, so the empty set is the fact that contains nothing. Absolute nothing is impossible (contradictory) because when there is nothing, there must necessarily be the fact of nothing, which is something, the empty set. The empty set then necessarily forms the set (fact) that contains the empty set, and thus a whole set-theoretic world is necessarily generated from the empty set (which is necessarily generated from nothing), and this world constitutes the mathematical world – all mathematical structures are constructed from sets – see e.g.

http://www.transcurve.net/profiles/blogs/nothingness-mathematics-starts-with-an-empty-set

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Inverse-square abstraction extends infinitely, but as you know few physicists buy into the empirical existence of anything infinite. And of course you insist in limiting your discussion in order not to have to deal with harder cases of mathematical structures, like irrational or imaginary numbers. And you keep being silent about logic…

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