Defending scientism: mathematics is a part of science

1+12[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.

341 thoughts on “Defending scientism: mathematics is a part of science

  1. I find it frustrating that critics of scientism constantly get accused of “territorialism,” while defenders of it don’t get labeled with the opposite term: cultural imperialism.

    The thing is – I’m a critic of scientism. Were I to embark upon a project in the field of art criticism, I would first spend a lot of time becoming familiar with the history, scope, and important figures in the field. Then I would focus upon those lines of discussion that most aligned with (or opposed) my specific interests. What I wouldn’t do is claim that the way our brains work is irrelevant. In fact, given my interests, I’d be looking at how it fit in or didn’t fit in to existing frameworks.

    By the same token, if someone were to claim that, say, Frege’s ideas were irrelevant to a discussion of whether mathematics is empirical, or that to have the discussion at that level of analysis was just wrongheaded and that we need a purely reductive, scientistic explanation only, I’d say that such a person was being territorial. Or culturally imperialistic, I guess, if I were feeling particularly dramatic.

    The thing is, there’s no easy way to say what’s relevant or what’s a valuable level of explanation. It’s going to be messy and overlapping. What we lack in these discussions is even an acknowledgment of the fact that we’re not even talking about the same category of things most of the time. Thus my recent slew of comments trying to ponder what’s going wrong.

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  2. Hi Aravis,

    . The point, however, is that *when we engage in art critical discourse* the brain level of explanation is not the *relevant* one, given our interests in engaging in art critical inquiry.

    Sure, agreed. It’s not the most relevant one (the word “everything” in my previous statement was about then going on to explain emotions in terms of brain states, etc — I agree that that’s not what literary criticism is about).

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  3. Hi labnut,

    The problem is that you cannot conceive of the mind creating something that is independent of the brain’s neurones.

    You’re right that I can’t! I readily accept (as I’ve stated plenty of times) emergent phenomena at a higher-level of description. However these are not “independent” of the lower level, but are products of the lower level.

    Note, by the way, that your code is not a product of the computer that it then runs on, so the analogy is not so good with a brain, where the higher-level behaviour really is a product of the lower-level hardware (in a neural network the “software” is the result of the hardware configuration).

    And, again, note the word “everything” in my above claim.

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  4. Just as an exercise, why stop at the brain level? Why not come up with a quantum mechanical theory of art criticism? You see how silly that sounds? And yet, our brains are made of quarks after all…

    Same reason you don’t want a quantum mechanical theory in neurology – because the effects that occur at the quantum mechanical level largely disappear (or are causally irrelevant) at the level being discussed.

    The question is whether it’s *relevant* to the problems you’re trying to solve, or the issues you’re trying to get at. So for some theories in art criticism, the brain probably *is* irrelevant.

    Some of what’s going on here is that something like a standard theory of reference isn’t satisfying to some, because it seems to hit a dead end with “inscrutable”, “fundamental” concepts. There’s no reason not to go at it from a different angle.

    But there’s no reason a theory of art criticism has to involve *only* the brain, or *only* a scientific explanation.

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  5. Coel,
    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.

    I think the discussion has missed a really fundamental and fatal flaw in your argument. The problem is contained in your bland and expansive term ‘knowledge about the universe‘. In the science context, what does that mean? Knowledge could be an itemised list of measurements and observational data. But that is not what we really mean.

    What we really mean is contained in a quite foundational assumption that underlies all of science. That assumption is that there exists an underlying order and regularity that we call the laws of nature and that ultimately we can express these laws of nature in a mathematical form.

    Science, then, is a process of discovering and verifying the laws of nature. Observational data are the clues we use to uncover the laws of nature. What are the laws of nature? There is the prescriptive and the descriptive view, but for the purposes of my argument, we need not worry about the difference.

    To test whether something is science we ask the following questions:
    1) does it have raw material in the form of observational data?
    2) can the raw material be used to reveal laws of nature(or at least refine our knowledge of them)?
    3) can we empirically verify our conclusions?

    The claim that mathematics is science fails on all three points, because:
    1) the raw material of mathematics are previous conclusions and logic;
    2) it does not reveal laws of nature;
    3) the conclusions cannot be empirically verified, they are verified by the application of logic only.

    Crucially, mathematics is independent of the laws of nature and tells us nothing about the laws of nature. Thus it cannot be science. You have confused the tool with the object. To a plumber every tool looks like a wrench. Scientismists are the plumbers of science.

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  6. Indeed, far from being territorial, it actually represents a much deeper understanding of the nature and purpose of inquiry

    I’ve got a deep understanding of the nature of inquiry. My deep understanding is that I want to know stuff. I want to know what an “intention” is and how it came to be. I want to know why language runs into so many problems when it tries to explain things. I want to know if moral judgements have a connection to reality or not, and what such a connection would look like.

    If finding out that stuff requires a butter knife and a prostitute from Milan, then I’ll buy some silverware and condoms and move to Italy.

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  7. Hi labnut,

    That assumption [that underlines all of science] is that there exists an underlying order and regularity that we call the laws of nature and that ultimately we can express these laws of nature in a mathematical form.

    I don’t see that as an assumption, but as a conclusion of empirical observation. We can see and test whether “laws of nature” do work and we can test whether expressing the laws in mathematical form works.

    The claim that mathematics is science fails on all three points, because:
    1) the raw material of mathematics are previous conclusions and logic;
    2) it does not reveal laws of nature;

    Again, I would disagree. I see the “raw material” of mathematics as noticing regularities in nature and codifying them into axioms. The axioms are “laws of nature”. Where else do you think the axioms came from? Why did Peano arrive at the particular axioms that he did? My answer is that he arrived at them because they model the world (= are “laws of nature”), having arrived at them by distilling maths that had previously been arrived at in the real world, starting from a farmer counting sheep and that sort of thing.

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  8. Hi Thomas,

    Seriously . . . “to understand the power of the poem” we’d have to do all that?

    Yes, to understand the power of a poem you do indeed have to understand how it interacts with human brains! It is, after all, human brains that do emotion and feeling, and the power of a poem is all about human emotion and feeling. Take the humans out of the picture and the poem would have no “power”. Note that I did not say that you had to understand everything about the brain in order to make a start on that.

    You will perhaps excuse those of us who reject this “one size fits all” epistemic approach.

    That’s fine, but go ahead and demonstrate that some “other way of knowing”, distinct from science, does actually work.

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  9. I think it is just a wee bit presumptuous to conclude that poets aren’t in the business of seeking knowledge. A poet is able to explain the natural world in ways a scientist never could.

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  10. Hi Paul,

    Is a tautology true in virtue of working better in the world or true whether or not it works better in the world?

    I’d go for the former. I’m not even sure what “true” means in the phrase “true whether or not it works better in the world”. Can Newton’s laws be “true whether or not they work better in the world”?

    do you hold that the following common definition of a tautology is wrong?: A tautology is a statement that is true whatever the truth-values of its component statements.

    Hmm, let’s think. Doing a substitution:

    “A tautological statement is one that {works in the world} regardless of whether the component statements {work in the world}”.

    That seems ok to me, so no I don’t hold that it is wrong.

    This definition doesn’t link the truth of tautologies to working better in the world after all.

    Err, it seems to me that it does!

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  11. Hi schlafly,

    The [science and engineering] professors act as if a proof is some sort of mysticism or voodoo with no applicability. Most of them do not understand what a proof is.

    You seem a bit jaundiced against science professors! I hardly think that your statement about them is fair.

    If so, I am likely to get 2.01 or something else not exactly 2. The mathematician says that 1+1 is exactly 2. So what have you tested?

    It’s exactly the same in physics! A physical law (expressed mathematically) will give an exact relation. A real-world measurement will have some error, and so the relation will not be exact. For example if the “law of physics” says that gravity obeys an inverse-square law, then actual measurements will deviate around that exactly as you describe. So again this supposed distinction between maths and physics doesn’t hold up. In both cases maths and physical laws are abstractions, idealised models that are adopted because they are useful.

    Aravis,

    Coel has no interest in how the practitioners of other disciplines do what they do. He has no interest in the concepts they use or how they define and understand them. And more importantly, he doesn’t think he needs to have any such interests.

    You seem a teensy bit jaundiced also! And equally unfair. Thinking that different disciplines should mesh with each other — rather than being compartmentalised and ignoring each other — is the exact opposite of having no interest in concepts in other disciplines!

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  12. Hi labnut,

    There is knowledge in poetry, but poetry is not only about knowledge, it’s also about things like communication and invoking emotion. That’s why a simplistic “poetry is science” is not mandated under my definition.

    Hi Massimo,

    I find it frustrating that critics of scientism constantly get accused of “territorialism,” while defenders of it don’t get labeled with the opposite term: cultural imperialism.

    Oh but they do! Just google “scientism” for a whole lot of anti-scientism sentiment, much of it along that line.

    I just finished a first draft of a new SciSal essay on the rise of “radical empiricism.” Stay tuned, probably out by the middle of next week…

    Looking forward to it!

    The point is that it isn’t explanatory. Just as an exercise, why stop at the brain level? Why not come up with a quantum mechanical theory of art criticism? You see how silly that sounds? And yet, our brains are made of quarks after all…

    As I see it, scientism is not a denial of emergent phenomena, nor a denial of the appropriateness of different levels of explanation, it is simply the idea that the different levels need to mesh with each other seamlessly, and not be treated as compartmentalised and independent.

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  13. Coel, you say that math axioms are codified regularities of nature. The most common axiom system for math is ZFC (Zermelo-Frankel). Can you explain how those axioms relate to nature?

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  14. Coel, you repeatedly deny any distinction between a definition, a theorem, and an equation that empirically seems approximately valid. So I would lump you in with those other science and engineering professors who do not recognize the value of a proof.

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  15. Coel,

    I’m totally mystified by your attitude towards mathematics.
    As schafly remarks: what about the axioms of set theory (ZFC)?
    They were invented to avoid mathematical inconsistencies, with the stress on the word mathematical. They weren’t invented to avoid physical problems.
    At least two of the axioms of ZFC don’t seem to describe the real world, if you ask me.
    1) The axiom of infinity, which basically says that infinite sets exist. If axioms were based on “the real world”, we would have had an axiom of finity.
    2) The axiom of choice. It leads to very strange consequence that don’t describe “the real world”. (Sets without measure, Banach-Tarski etc.)

    Nobody denies that with ZFC one can do maths that is useful to describe the real world. But strange enough, one needs for ZFC axioms that don’t describe the real world.

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  16. Coel:

    “Yes, to understand the power of a poem you do indeed have to understand how it interacts with human brains! It is, after all, human brains that do emotion and feeling, and the power of a poem is all about human emotion and feeling.”

    Well, thanks for the “human” connection, professor. I have to ask your pardon for indulging in some sarcasm: “the power of a poem is *all* about human emotion and feeling.” Is this an essay on what you did during your latest summer vacation? You can do better than this, can’t you?

    Frankly, some of us think you are trying to nail jelly to a tree,

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  17. For the record, I think mathematics is a science. (It’s the scientific study of a particular type of physical things), but the arts (like poetry, painting, literature, sculpture, and music) are not sciences (even though they involve the manipulation of physical things). I don’t think that ethics is a science either. (It seems like mostly guesswork and hope.)

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  18. The platform is different. I can filter or delete, but apparently not the user. But this platform has so many advantages compared to Blogger.

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  19. Is zero, or the empty set, a real mathematical object? If so, where is it represented in reality, except as an idea in relation to other sets? And if all numbers are defined in relation to the empty set, as in ZFC, and zero is not something which is represented in reality, then how are the other numbers actually represented? Isn’t it more accurate to say that numbers are ideas which provide useful correlations to real things?

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  20. Coel,
    I don’t see that as an assumption, but as a conclusion of empirical observation.

    Trivial objection which has no bearing on the substance of my arguments. Assumption or not, it does not change my argument in the slightest.

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  21. Coel,
    I see the “raw material” of mathematics as noticing regularities in nature and codifying them into axioms.

    Once again, you are ignoring the substance of my argument.

    What you say may be true in a limited way at the base of the vast scaffolding that represents mathematics, where the legs of the scaffolding touch the empirical world. Further up the scaffolding the raw materials are logic and previous conclusions, not observational data. All contact with the empirical world is lost as you go up the scaffolding.

    The operations on this raw material in no way reveals laws of nature. They merely create further logical relationships.

    These logical relationships are not empirically verified, they are verified by the application of logic only.

    Thus, once you leave the base layer of the scaffolding of mathematics, it fails all three tests for being a science, namely:

    1) does it have raw material in the form of observational data?
    2) can the raw material be used to reveal laws of nature(or at least refine our knowledge of them)?
    3) can we empirically verify our conclusions?

    The claim that mathematics is science fails on all three points, because:
    1) the raw material of mathematics are previous conclusions and logic;
    2) it does not reveal laws of nature;
    3) the conclusions cannot be empirically verified, they are verified by the application of logic only.

    Crucially, mathematics is independent of the laws of nature and tells us nothing about the laws of nature. Thus it cannot be science.

    Your arguments may be true in a limited and trivial way at the base of the scaffolding of mathematics but they are plainly untrue for the upper levels of the scaffolding. And the upper levels of the scaffolding represent most of mathematics as it is practised.

    Your arguments that mathematics is a science simply do not withstand the simplest tests for what science is.

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  22. Coel,
    your determined insistence that mathematics is science, in defiance of informed understanding and cogent counter-arguments, raises questions about your motivation.

    Massimo called it ‘cultural imperialism‘ while I prefer the term ‘empirical imperialism‘ because your beliefs have nothing to do with culture.

    Whichever term we use, it is a form of imperialism. This is characterised by:

    1) An immutable belief in the superiority of one’s world view.
    2) A disregard for the perspectives and insights of other world views.
    3) A need to impose one’s world view on others, displacing their world views.

    This is characteristic of scientism, which is a form of empirical imperialism. What is remarkable about empirical imperialism is how tone deaf and colour blind it is. Empirical imperialism is a harmful phenomenon because it insists on the hegemony of one world view to the exclusion of others.

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  23. Hi Coel,

    Build two models (or sets of models). Adopt the axioms of identity and non-contradiction in one model but not the other. Now test out how the models do for explanatory power, predictive power and parsimony. I would suggest that any model not adopting those axioms is going to be pretty useless in modelling the world.

    No, it still does not work,

    It would not be true that one model had more explanatory power, predictive power or parsimony unless it was also true that explanatory and predictive power is not absence of explanatory and predictive power and vice versa, or that parsimony is not unparsimony and vice versa.

    So your comparison assumes the very axioms it is supposed to be testing.

    We could continue this to any iteration you like but the same thing would be true.

    You cannot fill that broken straight even if you hold up infinitely many fingers to the dealer.

    If you disagree then you have to demonstrate it without assuming the axioms of identity and contradiction.

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  24. To (at least some) physicalists, there are no nonphysical abstractions in mathematics.

    * L. E. Szabó, Formal Systems as Physical Objects: A Physicalist Account of Mathematical Truth, International Studies in the Philosophy of Science, 17 (2003) 117.
    * L. E. Szabó, Mathematical facts in a physicalist ontology, Parallel Processing Letters, 22 (2012) 1240009 (12 pages), DOI: 10.1142/S0129626412400099.

    via http://phil.elte.hu/leszabo/Godel/2014-2015-1/

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  25. HI Coel,

    By “mathematical truth” I presume you are talking about propositions of the form “if {axioms} then {theorem}”. I can then ask you how you know that that statement is “true”. You could reply with a lot of symbolic logic. I could then ask how you knew that that {lot of symbolic logic} was true. I could keep asking at each reply how you knew it was true, and your reply could only be more symbolic logic.

    Not at all. My reply would be to ask what you could possibly mean by that question.

    With something like ZFC, you can ask about its completeness or its consistency, but is it true or false? That seems to be missing the point. It is neither true nor false, the same way that a Turing Machine is neither true nor false..

    But you can make a true statement about it. For example the statement “The Banach-Tarski Theorem can be proved in ZFC + The Axiom of Choice” is true, it is a fact.

    As for “useful and works in the real world” I am not sure that “useful” has ever been considered a criteron for truth, many people have found lies very useful throughout history.

    Also, what do you consider the “real world” for the purposes of that definition?

    Personally I think that mathematics is part of the real world.

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  26. When you talk about interdependence, I have already agreed that there is interdependence in the context of discovery for all three. The difference I was pointing out was in how claims are justified and the expectations of the justification. I hold that there is a useful distinction to be made here.

    There is also the difference in what is being inquired into. Mathematicians (pure) make claims about formal systems and not about the real world. However, I see you have already replied to that by saying mathematicians do make claims about the real world. Could you clarify what you mean by this by first explaining what you mean by real world and secondly how mathematics makes claims about that world.

    By saying there is no distinction between science and mathematics, are you saying there is no mathematical claim which is not about the real world?

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  27. Coel,
    In your conclusion you stated
    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.

    No, wrong. The kindest thing I can say about your conclusion is that it is a flat earth society view of knowledge. You have a model which consists of a substrate of observable reality and layered onto that substrate is knowledge derived directly from that substrate. If that were necessarily true there would indeed be no other ways of knowing. Unfortunately for you, your model is only true of science itself. It is not true of other domains of knowledge, as you claim. You are projecting your view of science onto other domains of knowledge, and that is why I earlier said that scientism is tone deaf and colour blind. It simply is blind to other ways of knowing. That is why you claim there are no other ways of knowing, but it is a false claim.

    Let me explain. When our brain was blessed with language it acquired a fundamentally new ability, the ability to create layers of knowledge, one built upon another. Our primate cousins could only ever deal with the tangible reality that directly presented itself to their senses. They were trapped by the confines of the empirical world, just as our other cousins, the scientismists are today. Scientismists are nothing more than highly evolved chimpanzees because they cannot see above the empirical world that traps them. What makes us different is that we can construct scaffolds of knowledge, with layer building upon layer. The scaffolds of knowledge free us from the shallow confines of the empirical, allowing us to gain greater insights and understanding. As we constructed these scaffolds of knowledge, speciation took place, allowing each scaffold to develop its own distinctive vocabulary, methodology and understanding. The results of this speciation was that knowledge was no longer seamless, as you claim.

    The scaffolds of knowledge freed us from the narrow confines of the empirical and this made possible the great flowering of our mind as it developed vibrant culture. It made ethical knowledge possible and it made aesthetic knowledge possible, outcomes that could never be derived directly from the sterile substrate called science. Our mathematics, literature, philosophy, ethics, the other arts, the humanities, are all outcomes of the scaffolding of knowledge. They can no longer be tested and understood using the tools of science because they exist at higher levels of understanding on the scaffolds of knowledge.

    Now you want to claim that all knowledge is empirical. You can only make that claim if you confine your view to the bottom layer of the scaffolds of knowledge, where they touch the tangible world. Look up at the scaffolds rising above empirical reality and you will see a new vista, reflecting the richness and power of the human mind. Then you will discover there are other ways of knowing. And then you will understand Siegfried Sassoon’s poetry.

    By all means, confine your view to the bottom layer if that is what satisfies you but then you will be little more than a highly evolved chimpanzee, a scientismist. The rest of us will be climbing the rungs of knowledge on our chosen scaffolds.

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  28. Hi labnut,

    What you say may be true in a limited way at the base of the vast scaffolding that represents mathematics, where the legs of the scaffolding touch the empirical world. Further up the scaffolding the raw materials are logic and previous conclusions, not observational data. All contact with the empirical world is lost as you go up the scaffolding.

    That “all contact is lost” claim is simply false. Suppose I start with an empirical statement: Jack has an apple. And suppose I have another empirical statement: Jack has an orange.

    Now let me apply some reasoning to those statements and arrive at “Jack has both an apple and an orange”. That last statement was not directly empirical, instead it followed from reasoning from the two empirical statements.

    Has that statement now “lost contact with the empirical world”? No it has not. It is *true* about the empirical world. That follows from (1) the axioms used being about the empirical world, and (2) that statement following from the axioms.

    Maths *is* *indeed* a whole edifice of scaffolding produced by reasoning from axioms. But, if the axioms are rooted in reality — which you have accepted — then the whole edifice is still about the empirical world and statements in it are true about the empirical world.

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  29. Hi Robin,

    If you disagree then you have to demonstrate it without assuming the axioms of identity and contradiction.

    We simply ask the question. If we assume axioms of identity and non-contradiction, can we get explanatory and predictive power? (answer, yes). If we do not assume the axioms of identity and non-contradiction, can we get explanatory and predictive power? (answer, no).

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  30. Been out of town, and following these stems is a nightmare.

    Robin Herbert, My initial question was not asking which? The addition “or mathematical axiom” as added as alternative phrasing. The answer to the question, duly given in the next paragraph, is “Neither!” The evidence from experiment is that there was a time when the human mind did not grasp object permanence (the axiom of identity,) and learned it from experience. There is no indication that the axiom of identity was used to analyze experience. Experiments show that animals can count some too and they surely aren’t using abstract mathematics to organize a bunch of sensations. You seemed to think you’ve a point by announcing you don’t remember moving your bowels as an infant. I have no idea how you’ve confused yourself into thinking this shows a priori intuitions (mathematical axioms) are required to organize experience. Bowel movements came first, then bowel control::Experiencing object permanence came first, then incorporating the generalization as an automatic assumption.

    It would be uncharitable to conclude you don’t even understand the objection, so I must conclude your real point is that philosophy ignores mundane human experience and scientific experiments. Point noted.

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  31. Out of town, and following those little lines is very difficult.

    In case there’s really a misunderstanding by C Lqrvy…

    “…you just don’t understand the nature of counting…” Well yes, it did and does take experience to realize that 3+7=7+3. Admittedly we internalize this very early in our lives, many of us before any formal schooling. On the general question of the empiricism of mathematics, the fundamentally empirical origin of its basic intuitions has been, however inadvertently conceded. Further, the crassly empirical nature of applied mathematics has been, unsurprisingly, neither contested nor acknowledged. The omission of applied mathematics on no reasonable grounds means those insisting on the non-empiricism of mathematical knowledge have presented partial, which is to say, misleading, evidence. The last resort, the claim that further mathematical truths (including, perhaps especially the mathematical constructions that cannot be empirically demonstrated) are not discovered using the empirical methods of science, is not really contested by anyone. But I guess the real point is that not seeing how there could be a separate existence of counting in a mathematical realm (or “conceptual space”) is a kind of stupidity but accepting this is good philosophy. I must plead guilty.

    I don’t think the big problem for the anti-science is whether the fundamentally empirical origins even of pure math, along with the obvious empirical nature of applied math, aren’t sufficient to justify seeing mathematics as a part of the science way of knowing the world. The big problem for them is explaining what it means to say mathematical knowledge is. I mean, if the Banach-Tarski theorem says a sphere is an infinite collection of points and we can use the axiom of choice to make the same sphere into two identical spheres without adding anything, what do we know? That the a priori analytical truth that a sphere is a volume of a certain shape is wrong? I submit that “knowing” stuff like that from the counterfactual parts of mathematics is much more like “knowing” what a second cousin twice removed is. Like “knowing” the what the fourth December payment on a compound interest loan is. LIke “knowing ” the difference between malfeasance, misfeasance and nonfeasance. Like “knowing” the legal grounds for annulment of marriage. LIke “knowing” when the king has been checkmated at chess.

    “…alternatively, that one needs to beef-up one’s first principles…” Perhaps I should have elaborated to be clearer. By indisputable, I meant “not permitting of argument, simple, mechanical, algorithmic, needing no further discussion, direct, straightforward, all the thinking is over.” So yeah, needing to beef up the axioms is again conceding the point. As I said, you might prefer to say that the views refuted are naive. But the news that mathematics is a creative endeavor is not universally understood. Indeed, it is commonly the understanding that mathematics’ conclusions are the cut and dried, black and white version of THE TRUTH about the world, and all of THE TRUTH, encrypted in the axioms of mathematics, from which THE TRUTH can be deduced, deduction being the only way of finding true knowledge.(Thoughts on the nature of the encrypter may vary!) Unless your point is that deduction is in fact the only way of deriving genuine knowledge, even when we have to choose the axioms we deduce from, by some undeclared standard? Given the difficulties that mathematicians had accepting negative numbers, infinitesimals, non-Euclidean geometry and infinites, this is not a straw man. I don’t understand how it is objectionable to cite incompleteness theorems as proof it’s just not so.

    “…I don’t understand what you’re saying..” It was awfully abbreviated, but I don’t like writing that much. Sorry. The thing about Platonism, mathematical Platonism and other variations is that if mathematical forms are prior to existing things, then there must be some way to match the mathematical forms to things. It needn’t be one to one but things do have to line up. “As above, so below” can’t be true if there are things below that don’t have some sort of counterpart above. The truths unproveable from any set of axioms rich enough to match to the maya, the phenomenal world, the prison of souls or whatever, would have no match in the mathematical empyrean. In effect they would appear out of nowhere. Thus, I think it sensible to conclude that even on its own terms, all forms of Platonism, mathematical Platonism and variations thereof are inconsistent. I suppose you could hypothesize a God to pick and choose amongs the Forms/Axioms but I didn’t want to get into theology. Since the real world has numerical values associated with it, I should think Platonism, mathematical Platonism and variations thereof also need to account for calculability, which also has its own difficulties Dr. Church tells us. Unless you really are objecting to ruling out Platonism etc.? [Yes, I’m afraid I think Max Tegmark is “variations” thereof and is already wrong, even without reading him. My bad.]

    “…The natural numbers are not definable in the theory of real-closed fields, which is why the theory of real numbers can be complete even though arithmetic isn’t…” This was not a comment on Euclidean geometry. But the popular view of mathematics’ superiority to science in providing knowledge does come from the knowledge that everything is implicit in the axioms, the unknown being undiscovered deductions. And that these axioms are complete in the sense Euclidean geometry is complete. And that these axioms and their deductions provide equally complete knowledge of the world as a logical a priori.
    I don’t believe any of these things are true, and until your correction thought mathematics did not claim them to be true

    As to the number line’s consistency, that was mainly humorous. But, I was thinking that if you started to slice up the number line, counting the integers, then slice up another one, counting up the real numbers, then matching the counts on the slices to each other, then it should be pretty straightforward to show that the counts of the counts don’t match up and that in a certain sense there is more counting for the real numbers than the integers. And that you can show there aren’t any way to slice up the number line that gives a count in between that of the integers and the real numbers. After all numbers are numbers, right? It’s not like the number line would have any gaps in it, that we might have to label with letters, cited in peculiarly unpredictable way, some here, some there. After all, numbers are numbers, right?

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  32. Going through the subsequent comments after coming back from out of town, I keep seeing the effort to redefine scientism as reductionism, which it is not, thus a repeated straw man fallacy. I also see persistent efforts to assume that pure mathematics is all there is, and mathematics about impossible objects is simply “knowledge,” end of story. I don’t see any serious justifications offered for either assumption. What I don’t see is any discussion of what mathematical knowledge is and how it compares with other kinds of knowledge, especially non-empirical forms of learning. The collective nature of scientific knowledge is barely alluded to!

    There is an appearance of the humanities in the thread as well. Most of it is a waste since it simply pursues the most ludicrous form of reductionism possible. Frankly, I think a certain amount of crude reductionism should be used in the humanities, starting with studying all example instead of just those deemed by authority of some sort to be worthy of study. In another vein, taking into account physical differences or processes, some as simple as blood sugar levels, seem to be an inescapable part of the esthetic experience. Lower levels of reductionism, such as a concern with economics and politics and sociology seem to me to be wholly inseparable from appreciating the communication of feelings and ideas in the various arts. I do not think, for example, the vicissitudes of tragedy from the days of Aeschylus to HBO can be deemed to be independent, following its internal logic, with crass empirical reality an irrelevance.

    In terms of feelings, science is about satisfying one particular feeling, curiosity about how the way things really are. Humanities in general seem to be about satisfying other kinds of feelings. If someone can really convince themselves that thinking science is the only effective way we’ve found to learn about nature; that we can also use scientific methods and its use of experience as the criterion of knowledge to study human nature and society and history as well; that we have found that the universe is in a sense lawful and that it is objective and independent of human minds…if someone has truly convinced themselves that thinking this somehow means you cannot understand Siegfried Sassoon’s poetry, I can only say that some unexamined impulse is leading to rationalization rather than reason.

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  33. There’s a lot in there that I find incomprehensible, so I’ll try to stick to a few limited points:

    >>Further, the crassly empirical nature of applied mathematics has been, unsurprisingly, neither contested nor acknowledged.

    Applied mathematics isn’t empirical. The problems applied mathematicians work on are *motivated* by problems that arise in empirical sciences. But the results that an applied mathematician discovers aren’t themselves empirical. A good example would be mathematicians who work on algorithms. An applied mathematician might be motivated to find optimal factorization algorithms for a certain class of matrices that typically arise in physics and engineering. The fact that those kinds of matrices arise in physics and engineering is an empirical fact. But when the applied mathematician shows that one kind of algorithm is more efficient than another, or that one is subject to rounding error more than the other, that is a purely mathematical fact, not an empirical one.

    >>Unless your point is that deduction is in fact the only way of deriving genuine knowledge, even when we have to choose the axioms we deduce from, by some undeclared standard? Given the difficulties that mathematicians had accepting negative numbers, infinitesimals, non-Euclidean geometry and infinites, this is not a straw man. I don’t understand how it is objectionable to cite incompleteness theorems as proof it’s just not so.

    It’s objectionable because the incompleteness theorems don’t show what you think they show. To be clear, I’m not saying you’re necessarily *wrong*. I’m just saying that the incompleteness theorems are irrelevant to this particular issue.

    >>But the popular view of mathematics’ superiority to science in providing knowledge does come from the knowledge that everything is implicit in the axioms, the unknown being undiscovered deductions. And that these axioms are complete in the sense Euclidean geometry is complete. And that these axioms and their deductions provide equally complete knowledge of the world as a logical a priori.

    >>And that you can show there aren’t any way to slice up the number line that gives a count in between that of the integers and the real numbers.

    What? Are you saying you’ve settled the continuum hypothesis?

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  34. >>With something like ZFC, you can ask about its completeness or its consistency, but is it true or false? That seems to be missing the point. It is neither true nor false, the same way that a Turing Machine is neither true nor false..

    Actually, this is a borderline incoherent view (and this IS where Godel’s theorems are relevant.) When you say “you can ask about [ZFC’s] completeness or its consistency”, it sounds like you’re saying the sentence “ZFC is consistent” is either true or false. But a lesson from Godel’s theorems is that the sentence “ZFC is consistent” is essentially just a complicated arithmetical statement (call it S.) So if you think “ZFC is consistent” is either false, then you’re committed to thinking S is true or false. But now that leaves you in the position of having to explain why some arithmetical statements can be true or false, but not others. Moreover, you have to explain why some arithmetical statements can be true or false, but not statements about sets.

    The view you’re espousing (“you can ask about ZFC’s completeness or consistency, but…it is neither true nor false”) sounds like a version of what Hilbert was going for, but his program famously came crashing down.

    questionYou seem to think there is an answer to the question whether ZFC is complete or consistent. That is, you think the sentence “ZFC is consistent” is either true or false. But part of the upshot of Godel’s theorems is that the sentence “ZFC is consistent” is essentially equivalent to asking a (very complicated) arithmetical question (

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  35. Coel Hellier: ” … I have argued that all human knowledge is empirical and that there are no “other ways of knowing.”

    I must disagree to this 100%.

    First, there are two types of ‘empirical’, the universal and the ‘theory-based’. The realities of human intelligence and human consciousness are universally empirical perceivable, and this empirical knowledge will last forever as long as there are humans. The so-called Higgs-boson is 100% theory-based, and it can be fallible in 20 years when a ‘new-physics’ is discovered. So, what kind of empirical you are talking about?

    Second, 1 + 1 = 10 is exactly correct in binary-number-system which is a symbolic ‘representation’, purely mathematical. There are two types of physics: the nature-physics (which governs this universe) and the human-physics (which is ‘discovered’ by human about the nature-physics). And this human-physics is only a ‘small’ subset of the nature physics. There are also two types of math: the nature-math and the human-math.

    It is a big issue about what the ‘Nature-math’ is. I will define it as having only three parts {zero, numbers (real and complex numbers), infinities}. While the human-math is ‘based’ on this nature-math, it has a big effort doing its own ‘constructions’. Thus, the Grassman numbers, Quaternion and Octonions numbers are parts of the human-math. With the nature-math as the basic-lego, the human-math is all about the ‘construction’ of ‘structures’.

    Third, the term of ‘science’ is not clearly defined. The science with the Popperianism will definitely exclude Math as a part of science. The science of Duhem-Quine type web of belief {… there is [thus] no theoretical fact or lawlike relation whose truth or falsity can be determined in isolation from the rest of the network. … to include logical (coherence), aesthetic (simplicity) and sociological considerations. …} can indeed encompass the Math as part of it.

    Your article is only an issue about the ‘human-physics vs human-math’, a linguistic issue. Yet, there is a true issue, that is, the nature-physics must be identical to (=) the nature-math.

    The nature-math has three parts (zero, numbers, infinities), and the (zero and infinities) are ‘timeless and immutable). The key principle in nature-math is the {principle of ‘unreachable’}. The prime-number is unreachable by ‘multiplication’ operation. The irrational numbers are unreachable by algebra, etc.

    The nature-physical (not physics) universe has no zero (nothingness) nor infinities. If nature-physics is identical to nature-math, the zero and infinities must be hiding beyond this nature-physical-universe, with the following process (the creation process).

    Nature-physics (base, with zero and infinities)  creation process {timeless process (producing a nature-constant), immutable process (producing an unchanging structure)}  nature-‘physical’-universe.

    I have discussed this nature-physics-processes many times and will not repeat them here. Although the {nature-physics = nature-math} is discussed first time here, it was discussed in detail in the book “Super Unified Theory”, and an abridged version is available at http://www.prequark.org/Mlaw.htm . Interestingly, this ‘Grand Unification of Mathematics and Physics’ issue now becomes a hot subject (see, Not Even Wrong, http://www.math.columbia.edu/~woit/wordpress/?p=7114 ).

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  36. “In terms of feelings, science is about satisfying one particular feeling, curiosity about how the way things really are. Humanities in general seem to be about satisfying other kinds of feelings.”
    Once you’ve made this concession, your argument collapses (or at least Coel’s does.)
    “In another vein, taking into account physical differences or processes, some as simple as blood sugar levels, seem to be an inescapable part of the esthetic experience.”
    Nobody doubts that this will tell us something about how the brain processes certain stimuli, including those labelled ‘aesthetic.’ But that will never tell us anything about aesthetics. The discourse concerning , say, poetry, is of a different kind and a different order than that concerning brain functioning.
    “What I don’t see is any discussion of what mathematical knowledge is and how it compares with other kinds of knowledge, especially non-empirical forms of learning.”
    I disagree, but allowing this for a moment, I would say that Coel has not supplied this either. His argument tends to change the value of terms mid-argument; some important terms are not precisely defined; and his main argument depends on an epistemological argument he hasn’t given.
    In a previous comment, Selfawarepatterns wrote: “I agree with him that mathematics rests on foundations that are empirically observable, even if most of what mathematicians do is work on logical consequences of those foundations. I disagree that math is science, mainly because I see science as the pursuit of reliable knowledge of reality, and not everything mathematicians work on pertain to reality.” I actually have no problem with this; but Coel clearly does. Coel’s argument is totalistic, it leaves no room for disagreement or some niche of differing discourse.
    “The collective nature of scientific knowledge is barely alluded to!”
    According to Coel all knowledge *is* science, so what would be the point of discussing this?

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  37. Mathematics does NOT count as science, and the reason it doesn’t is simple: Everything on one side of the “=” is identical in meaning to what is on the other side. It’s all tautologies, and tautologies aren’t derived from scientific investigation.

    Look, if someone says, “All swans are white,” then that’s biology. Australia was explored, and the statement was determined to be false. “All swans are white” and “Some swans are black” are both scientific claims, one now known to be wrong, but both scientific nonetheless. If manifesting the property of being white were a necessary condition for something to be a swan, then the discovery of the Australian birds wouldn’t have proved the first statement false. If whiteness had been contained within the definition of swan, then the first statement would have been a tautology. Science could not have affected it.

    If someone says, “All vertebrates have spines,” then that isn’t science.
    Unlike the swan statements, it’s a tautology. It’s a use of scientific terminology, but that’s just a linguistic characterization. The statement itself isn’t a making a scientific claim.
    If someone says, “All triangles have three sides,” then that isn’t science either.
    In precisely the same way, “1+1=2” isn’t science.

    In the above paper, Hellier* touches on this just prior to the conclusion. He says one argument against his thesis is that “we accept the claim that 1 + 1 = 2 because it must be true, it is the only logical possibility.” That’s a correct argument, and while I’d have framed it differently, it does account for why “1+1=2” isn’t science.

    So,let’s look at why he discounts this argument…
    “a critic will say, 1 + 1 equaling 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?”

    What has happened here, is IMO just a problem of getting tangled up in a version of “possible.” We go from possible to logical possibility to possible worlds..

    Ah! Now we’ve arrived at modal logic. The operators look like dingbats. Calgon, take me away!

    That’s an unnecessary detour, and it MIGHT be a dead end depending on a what assumptions one makes about a bunch of different things. The point is that it’s entirely unnecessary to get involved in logical proofs.

    “1+1” is “2” by definition. “1+1” is just another name for “2.”
    If the tangle of logical concepts were a Gordian knot,
    then that’s all it takes to cut through it.
    _____________________________
    *: I’ve noticed others saying ‘Coel,’ but I’m not sure of the etiquette and I didn’t want to presume.

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  38. This is the best post I’ve seen from you; and you are replying to Coel on his own terms.
    Coel said that “zero” was only a mechanism used to build models applicable to the universe; that cannot possibly be the case, The universe cannot contain anything just in and of itself equal to zero, and infinity goes beyond the testable in our knowledge of the universe. Clearly both terms have something to do with something other than the empirical. Zero and infinity are problematic and require considerable thought; but neither refers to the physical universe as we know it.

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  39. That is one of the points I’ve been trying to raise.
    Coel writes as though his position is news; in fact it necessitates knowledge of the history of debate on the same issues.

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  40. Coel,
    you keep writing about comparative models, but evade your responsibility; it is YOUR argument; thus YOU need to build the comparative models and show how one is less effective than another. None of us has to do anything until you make this case.
    But here’s the problem: even if you make that case it won’t prove your main argument. Nobody doubts the usefulness of mathematics in understanding the physical world; the question is whether this means that mathematics is necessarily an empirical science. All your modelling will not get you that.
    “X=X” is only true about the physical world in a trivial way. But as the Identity Principle, it governs the deployment of logical discourse, having nothing immediately to do with the physical world. It doesn’t model anything; it is simply a regulation over logical discourse itself.
    Again, you need to make an epistemological argument here; but your epistemology is only primitively empirical, and yet you think it answers everything.
    I strongly suggest reading in the history of your position, to clarify what arguments collapsed, and which remain sustainable.

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  41. Coel:

    I have enjoyed reading your blog and your comments here, however. I am not sure I agree with your that Mathematics is part of science. 

    I guess I am a Humean on this subject and think that there are two kinds of knowledge 1: relations of ideas (math, geometry, algebra) 2: matters of fact and existence (science).

    Below is some comments why I don’t agree with your view.

    1: Over generalisation.

    You cite one example 1+1=2 and claim that this is an empirical discovery. It is possible that someone might come to know that via experience, however your sample size is too small. Consider this 455+329 = 784. This is a true statement, however it is doubtful that this or nearly all other Mathematical truths are discovered empirically – that is by perception, testimony or cause and effect reasoning. Much the same can be said about algebra or geometry. In short, we know them without recourse to experience.

    Summary: your sample size is too small. Over generalisation. 

    2: Empiricism is not necessary. Confusing discovery with justification.

    We may discover mathematical truths via empiricism however this is distinct with the concept of justification. As I said in (1) most or nearly all truth statements  in mathematics are arrived at and justified via non-empirical means. If this is true then empiricism is not necessary for the discovery and justification for its claims. Therefore, mathematics appears, at this point, to be largely autonomous from empirical science. (more below see proviso) 

    3: Justification.

    Hume said roughly the following. Any matter of fact can be denied without contradiction: that the sun will not rise or flame will not give out heat is false but not contradictory. Any statement that is a relation of idea cannot be denied without contradiction: that a triangle has five sides is contradictory and a self-evident absurdity. 

    On the assumption that any contradictory statement cannot be true we can know that denials of true statements of mathematics, algebra etc are false without having to perform any empirical study.

    Why? Because, once we have the concept of a triangle or the concept of number and symbols we can, without any further experience (beyond that necessary to understand the concept) know that it is true – that it must be true, that its truth has a self-evident quality of certainty with it that can never be attained by any matter of fact (empirical statement) because it is always logically possible to assert the contrary. 

    Consider an example:

    All criminals have broken the law. 

    Once i understand the concepts involved I know that if X is a criminal then X has broken the law. This is logically and conceptually necessary and it is known a priori. I do not need to perform any empirical observation or test to know it.

    Furthermore, the truth of the statement that all “criminals have broken the law” does not depend on the existence of any non-abstract  entity such as a criminal or an abstract one (law) there may be no criminals and no laws but the statement is still true. 

    The upshot then, is that there are statements that purport to be knowledge that do not depend on empirical inquiry or empirical inquiry is not necessary for us to know that they are true. Consequently, mathematics and other subjects are not part of science. 

    4: Quine and a proviso.

    And yet and yet…. Might Coel lose the battle but win the war? 

    Quine claimed something to the effect that any statement may need to be revised in light of empirical discovery. For example logical laws (in light of QM) and Euclidean to  non-Euclidean geometry. Coel could claim that truths of mathematics etc might be falsifiable by new empirical discovery.

    He has said something to this effect. He says that our confidence in axioms is because they “work” or that they are coherent with our experience. 

    This is an interesting claim and worth further thought. What I would say, at lest, is that it allows the logical possibility that some (all?) true statements of relations of idea may be falsified or must be coherent with some empirical discovery. 

    We then have two things, the possibility of falsification. Secondly, the requirement that a necessary condition be stated that relations of ideas (math, geometry etc) not contradict any empirical finding. 

    If this be the case, then one cannot claim that math etc is a fully autonomous discipline from science and that there is some relation to it. This then is coherent with Coel’s claim that knowledge is all interconnected. However, despite this proviso, in practice and mostly in principle relations of ideas are independent in terms of justification. 

    Summary:

    1: Sample size is too small and you have made an over generalisation with your example of 1+1=2.

    2: You have confused the concept of discovery with justification. 

    3: The justification of mathematical truths is non-empirical because it depends upon concepts and their relations and not upon facts and matters of existence. Conceptual truths cannot be denied without contradiction which is what give us their certainty. This is not the case with matters of fact which can be denied without contradiction and depend upon observation and experience to know them. 

    4: The Proviso.

    1: Relations of ideas may be falsified.  
    2: empirical conditions are, possibly, necessary but not sufficient for the justification of relations of ideas (math, geometry etc.)

    Concluding comment. If the above is true, then you are not completely wrong, but would have to revise your claims (weaken them and re-define them) 

    Best 

    Michael.   

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  42. It occurs to me that the tautology claim and the axiomatic argument, although not identical, are related it’d be useful to get into how.

    “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.
    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.”

    if someone said that they believed “1+1=2” because of Peano’s axioms, I’d not believe them either! I like this. It’s kind of funny 🙂

    No, we believe “1+1=2” because we know what two means. 1+1 is essentially the definition of 2 as 1+1+1+1+1 is the definition of 5. We learn numbers from counting, and adding ‘1+1’ is the same as counting to two. Now I suppose you could argue that counting is science. I find the idea outlandish, but others may not. Hellier says something like that here:

    “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.”

    First, I’d say that the apple example is not a confirmation of ‘1 + 1 = 2,’ but an illustration of what ‘+’ means. ‘Apple and bag’ type stories don’t teach us that particular equations are true. Imagine how tired elementary school teachers would get by the time they got to 12+17! These are illustrations of what addition means just as we learn what subtraction means with stories of giving people things. It all reduces to meaning and therefore isn’t science. I think what Hellier says next makes that even more clear.

    “1+1=2… 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.”

    That’s wrong. We can observe all sorts of things falling, including apples. Hold a ball up in the air, and release it and it falls. This happens with everything our chubby little hands can grab… until we get a helium balloon! So, while ‘1+1’ and ‘2’ are merely a matter of understanding the meaning of something, seeing things fall and gravity are completely different. Falling is something humans have always known. Falling was known before humans existed. Gravity was discovered and explained falling. That’s science. Numbers were not discovered as gravity was. Numbers are a name for something exactly as falling is a name for something. That’s just not science. It’s language.

    Now, while. “1+1=2” is self-evidently a tautology, the obviousness is only because it’s so basic. More complex mathematical equations aren’t so obviously a case of definition and naming. A relatively simple equation like “√14400=120” bears no resemblance to counting, but it’s still a tautology. Because everyone who sees “√14400” knows that it’s identical in meaning to “120” (as opposed to √14401), it’s an equation where axioms (even if we don’t use that label) come into play. Maybe someone actually learned √144 by observations and possibly you did the same with √100, but no one did this for √14400. I don’t believe that any more than learning addition from Peano. No, we know the answer because we know the distributing rule for square roots, which I’d call an axiom.

    It’s with the more complex equations and the use of axioms that arithmetical tautologies have a different character than ordinary language tautologies – but they’re still both just language and independent of science. Math just gives us super useful rules that provide shortcuts so that we don’t need to look up definitions all the time – and using a dictionary because you don’t know the definition of horology is doing the same thing as using a table for math. This isn’t about science. It’s just language.

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  43. Tautologies aren’t true by virtue of what “works better.” A tautology is true by virtue of the meaning of the elements that compromise it. You can come to know a tautology is true either by virtue of definition or if it’s in an artificial language, you can learn it’s true because you apply some operation to something you know by definition.

    That’s why math and logic and basic vocabulary aren’t cases of scientific knowledge.

    If ‘Indianapolis is the capital of Indiana’ isn’t something we learn from science (& it isn’t), then none of this is science.

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  44. “this supposed distinction between maths and physics doesn’t hold up. In both cases maths and physical laws are abstractions, idealised models that are adopted because they are useful.”

    I get the impression that you might say anything adopted because it’s useful is science. If you give something a name, then you have adopted a useful practice. You needn’t go point to a thing if you name it. Names for properties are even more useful. Numbers are names for varieties of sets of things just as orange names the color of things. Math is indeed useful, but it’s language.

    Language is not science.
    Language precedes science.

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