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.

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341 replies

  1. but as you know few physicists buy into the empirical existence of anything infinite.

    Do I know that? Most cosmological models extend to infinity, and most physicists today would assume that the universe extends to infinity.

    Extension to infinity is a sensible abstraction if you have no reason for a boundary.

    like irrational or imaginary numbers.

    Irrational numbers such as pi, sqrt(2) and e are used all the time in physical models. They are just as much a part of physical models of reality as are the counting numbers, 1, 2, 3 ….

    Again, all of these derive from real-world experience. The first discussion of pi would have been in terms of the real-world circumference and the real-world diameter of a real-world circular object.

    As for imaginary numbers, they also occur all the time in physical models of reality.

    And you keep being silent about logic…

    Not really, my argument covers logic in exactly the same way as maths (as my article says). Both logic and maths are arrived at as abstractions of deep regularities in real-world experience.

    The reason I’ve not talked about logic more is that it is exactly the same argument.

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  2. Hi DM,

    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, …

    I wouldn’t quite use the phrasing of saying that the multidimensional mathematical topologies themselves “are empirical”, I’d say that they are (quoting you) “extrapolated from observation of the physical world” and thus that maths overall is founded in empiricism. The edifice of mathematical reasoning then constructed on that foundation is tethered to the empirical world by those tautological tethers.

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  3. Coel, I think you have largely left logic alone because your case would be even more difficult to make there. What sort of empirical facts justify, say, modal logic? Or paraconsistent logic? What kind of experiment would you devise to explore the possibilities offered by many-worlds logic?

    And every time you are pushed on the disconnect btw mathematical construct and reality you insist that such constructs are “extrapolations” from the real world, once again mistaking the (possible, you are speculating) origin of basic mathematical notions with the entirely logically distinct issue of the epistemology of mathematics. There are no parallel lines in the world, indeed, there are no lines period. And the parallel postulate cannot be tested empirically. In fact, the whole notion doesn’t seem to make much sense.

    You also keep coming back to analogies btw math and physics, like the inverse square law. But nobody here has denied that math is useful to science. It couldn’t be otherwise because the empirical world seems to be logically coherent and mathematically consistent, so it *has* to be the case that logic and math are useful to understand the world. But your notion is exactly backwards.

    Oh, and my understanding is that many (if not most) physicists reject the existence of actual infinities.

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  4. he is simply wrong in suggesting that my article was not about justification. All of it was about justification
    ———-

    I’ve already quoted the passages in which you clearly conflate the etiology of a belief with its justification.

    Justifications are normative, whereas mere reasons are not. In mathematics and logic, the justifications for the relevant beliefs are a priori, even if the beliefs were acquired by tying string around balls, counting, etc.

    That you continue to give the latter sort of account, when it is the former that is apropos demonstrates that you do not understand the difference. That you continue to double down, when this is pointed out to you, by one person after another, is too bad.

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  5. Hi Massimo,

    What sort of empirical facts justify, say, modal logic?

    Presumably you can predict my answer by now? Logic and maths are adopted as models of the world, because adopting the logic and maths gives greater explanatory and predictive power.

    Using modal logic quite obviously adds to the power and utility of models.

    Or paraconsistent logic? What kind of experiment would you devise to explore the possibilities offered by many-worlds logic?

    These extrapolations of logic are again attempts to add to the explanatory and predictive power of models. How well they succeed can be argued about, but that is the aim.

    As an example, electronic engineering often uses four-valued logic for real-world applications, where in addition to yes/no values you also have “don’t know” and “don’t care”. (Don’t know and don’t care also being concepts with real-world correspondence.)

    Paraconsistent logic is an attempt to deal with partial and inconsistent information, again a common real-world problem.

    you insist that such [mathematical] constructs are “extrapolations” from the real world, …

    They are. One arrives at both mathematical and physical models as abstractions of real-world behaviour, and once you have a model you can extrapolate from the model.

    There are no parallel lines in the world, indeed, there are no lines period.

    A line is an abstract concept, being the shortest distance between two points. That is just as much a physical concept as a mathematical one.

    Which is a more general issue. Nearly all of your objections apply just as much to physical concepts as to mathematical concepts, so are not relevant to distinguishing between maths and physics.

    And the parallel postulate cannot be tested empirically.

    Yes it can! All of the observational evidences of general relativity are tests for curved spacetime, which is equivalent to the parallel line postulate not holding.

    You also keep coming back to analogies btw math and physics, like the inverse square law.

    The main point there being that physical concepts have the same properties as the maths concepts you are pointing to (being abstractions, being concepts not objects, extending to infinity, etc).

    … so it *has* to be the case that logic and math are useful to understand the world.

    Not if the logical system of our-maths or our-logic were fundamentally different from the logic by which the real world operates. The fact that our-logic and our-maths matches the real world is because that’s where we got them from.

    … because the empirical world seems to be logically coherent and mathematically consistent, so it *has* to be the case …

    You’re saying that it *seems* to be the case that our universe follows our-logic and our-math, therefore it *has* to be the case that it does? How does that follow?

    Oh, and my understanding is that many (if not most) physicists reject the existence of actual infinities.

    Do you have quotes for that? As I say, most cosmologists today presume a universe extending to infinity (or, at least, so far beyond the observable horizon that they don’t have to ask the question, which is effectively infinity; the concept of infinity is very commonly used in physics).

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  6. Hi Aravis,

    Justifications are normative, …

    And I am saying that the justifications for mathematical axioms is that they reflect the “normative” standard of real-world behaviour.

    What is your justification for the axioms of maths?

    In mathematics and logic, the justifications for the relevant beliefs are a priori, …

    I don’t accept that, unless you are just taking the axioms as a given, and not asking where they came from or why those axioms are justified (which would be a rather incurious attitude).

    … even if the beliefs were acquired by tying string around balls, counting, etc. That you continue to give the latter sort of account, …

    Showing that the beliefs match real-world behaviour is indeed *justifying* those beliefs against the normative standard of real-world behaviour.

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  7. What a wonderful analogy Coel. Thanks again.

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  8. And I am saying that the justifications for mathematical axioms is that they reflect the “normative” standard of real-world behaviour.

    ——

    As I guessed: you do not understand the concept of normativity, as it applies to both justification and truth. And it’s a shame that you seem to be completely uninterested in learning things from others. These conversations would be a lot more interesting.

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  9. Some opinions about whether the universe is infinite are collected in this new paper: Astrosociology: Interwiews about an infinite universe. It ends with this quote: “Few questions in science are further away from the immediate needs of anyone than such a question as the infinity of the universe. The person tries to understand nature objectively, but coming to the deepest issues, he unexpectedly sees himself as in a mirror.”

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  10. Seems to me that the question of whether the universe is infinite (whatever that means) is distinct from that of whether there are incfinitely countable objects within it. I was referring to the latter, and it is my understanding that, for instance, solid state physicists really don’t like the idea at all (they also don’t like the use of mathematical singularities to derive phase transitions – come to think of it, yet another example that Coel’s approach cannot handle).

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  11. >>I agree that mathematicians are doing empirical science: they are discovering the objective world of mathematical structures…in any case they discover facts about objective reality.

    “Empirical” doesn’t just mean “pertaining to objective reality.” It specifically refers to the role of the senses/observation/experiment in justifying a claim. The argument here is whether mathematical claims somehow need evidence obtained by the senses to be justified; that is a separate question from whether mathematics “describes objective reality.”
    are somehow justified by sense experience.

    >>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)

    Whether the existence of a set “would lead to a contradiction” depends on one’s background axioms. Take ZFC+(not-CH); this theory takes ZFC and adds the assumption that there is a set S with cardinality between the natural numbers and real numbers. This theory doesn’t lead to a contradiction (unless ZFC itself is inconsistent), so according to your “self-evident assumption”, we should be able to infer that S exists. However, we can also take the theory ZFC+CH, which takes ZFC and adds the assumption that there is a set B that acts as a bijection between the natural numbers and the real numbers. Once again, this doesn’t lead to a contradiction, so your “self-evident assumption” allows us to infer that B exists. But the existence of B rules out the existence of S, and likewise, the existence of S rules out the existence of B. So your own “self-evident assumption” is itself inconsistent.

    >>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

    The process you describe only generates the finite sets. Even if you assume that every set generates a powerset, and the union of any collection of sets forms a set, you won’t be able to generate an infinite set without a separate, additional axiom stating that such a set exists.

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  12. Hi Aravis,

    As I guessed: you do not understand the concept of normativity, as it applies to both justification and truth.

    You are the one that brought normativity into the discussion. I am just explaining where the axioms of maths come from and why they are justified. They are justified as models of real-world behaviour.

    If you don’t agree, what is your justification for the axiom of maths?

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  13. He told you several times: coherence and the production of interesting results, regardless of whether such results have anything at all to do with the real world.

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  14. Hi Massimo,

    He told you several times: coherence and the production of interesting results, regardless of whether such results have anything at all to do with the real world.

    Well no, he (Aravis) has not given that rationale (at least not on this thread).

    On coherence: does anyone have a proof that our-logic and our-maths is the only possible such system that is coherent? If so, that is very interesting.

    If not, then the question remains, why *those* axioms, from among all the possible coherent systems?

    Justification in terms of the results being “interesting” is a concept worth examining. Since the same justification could be given for theology, it doesn’t seem to me a very strong justification. Anyhow, human interest would be increased by real-world relevance.

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  15. >>On coherence: does anyone have a proof that our-logic and our-maths is the only possible such system that is coherent? If so, that is very interesting.

    If not, then the question remains, why *those* axioms, from among all the possible coherent systems?

    Peter Koellner has numerous papers on this topic, one being called “Truth in Mathematics: The question of pluralism.”

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  16. I didn’t “bring normativity” into the context of the discussion. Normativity is an inherent component of justification and truth, as these concepts are understood in Epistemology and the Philosophy of Language, which are the primary disciplines charged with their study.

    Since this dimension of these concepts is obviously unfamiliar to you, I am happy to discuss them briefly. (I’ve got a pile of papers to grade)

    Truth is a normative concept in the following sense: If ‘P’ is true, then one of the implications is that one ought to believe that P.

    Justification is also inherently normative, insofar as it is defined in terms of truth: J is a justification for P, if J is a truth-reliable reason for believing P. One of the implications, then, of J being a justification for P, is that one ought to accept J.

    Some have tried to naturalize the normativity of justification and truth, by appeal to evolution: “The reason why one ought to believe true things, is because believing truths is advantageous, selectively.

    This, then, renders the obligation to believe what is a hypothetical imperative: ” If one wishes to survive and propagate one’s genes, one ought to believe things that are true.” The trouble with this route is that *either* one ultimately loses the relevant normativity — if one does not wish to survive, the “ought” does not apply, or one simply pushes it one step back — “Why ought one to survive and propagate one’s genes”?

    My own view is that normativity is both irreducible and ineliminable. Indeed, it is one of a number of crucial, explanatory concepts that are irreducible and ineliminable. Another is intentionality.

    For those of us who accept the fundamental disunity of the sciences and embrace a number of distinct, autonomous levels of explanation, this poses no problem. For those who still cling to the fantasy of the unity of the sciences, however, it is a serious problem and one for which no one has yet found a satisfactory solution.

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  17. Hi Aravis,

    Since this dimension of these concepts is obviously unfamiliar to you, I am happy to discuss them briefly.

    That really is most kind!

    Normativity is an inherent component of justification and truth, as these concepts are understood in Epistemology and the Philosophy of Language, which are the primary disciplines charged with their study.

    Well here we return to a primary way in which our viewpoints differ. You might regard philosophy as the “primary discipline” with regards to notions of “truth” and “justification”, but most scientists would regard those concepts from a scientific standpoint, and do not regard science as subservient to philosophy on such issues (sorry!).

    In the interests of avoiding too much antagonism between us, how about you not assume that all such concepts should be discussed as philosophy sees them on philosophy’s terms, and I not assume that they should be discussed only as science seems them, on science’s terms? That way we might better achieve what is presumably one of the points of Scientia Salon, of different disciplines talking to each other.

    Truth is a normative concept in the following sense: If ‘P’ is true, then one of the implications is that one ought to believe that P.

    This is a good example of philosophy and science seeing it differently. In science “truth” is simply descriptive, the word simply denotes a correspondence with reality. It is not normative in the sense that one “ought” to believe it.

    Indeed, to a scientist that seems a rather bizarre concept. As I see it an “ought” statement derives from some goal, from human feelings and desires. Some humans might indeed value truth, but that is their personal preference.

    One of the implications, then, of J being a justification for P, is that one ought to accept J.

    This is why I did not discuss this concept at all in my article, since from the point of view of science “justification” does not entail any such “ought”; in science a belief is “justified” if the evidence suggests that it matches reality. That’s all the word means to a scientist.

    The trouble with this route is that *either* one ultimately loses the relevant normativity — if one does not wish to survive, the “ought” does not apply, or one simply pushes it one step back …

    I agree entirely with your account of why that reply fails.

    My own view is that normativity is both irreducible and ineliminable. Indeed, it is one of a number of crucial, explanatory concepts that are irreducible and ineliminable. Another is intentionality.

    Whereas I’d just declare that there is no such normative element to “justification” or “truth”.

    I’m also highly dubious about there being any “irreducible and ineliminable” explanatory concepts (indeed simply declaring them irreducible and ineliminable seems to be the absence of any explanation).

    Your saying that about “intentionality” does, however, help me understand your stance on that concept, as discussed in previous exchanges.

    For those who still cling to the fantasy of the unity of the sciences, however, it is a serious problem and one for which no one has yet found a satisfactory solution.

    In what way is: “there is no normative element to “truth” or “justification” in the sense that one “ought” to believe it” not a straightforward and satisfactory solution? (Though of course humans may have preferences on the matter.) This seems to me an invented problem that doesn’t actually exist and so does not need a solution.

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  18. >>“Empirical” doesn’t just mean “pertaining to objective reality.” It specifically refers to the role of the senses/observation/experiment in justifying a claim.<>The process you describe only generates the finite sets. Even if you assume that every set generates a powerset, and the union of any collection of sets forms a set, you won’t be able to generate an infinite set without a separate, additional axiom stating that such a set exists.<<

    Why not? The number of sets that are generated grows without limit.

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  19. In what way is: “there is no normative element to “truth” or “justification” in the sense that one “ought” to believe it” not a straightforward and satisfactory solution?

    ———–

    It’s straightforward, but it’s not a solution.

    Truth and justification are treated as normative, by everyone, in their daily conduct, as well as in ordinary discourse and speech. We encourage people to believe true things, not false ones. Because of this, we encourage people to accept good reasons for believing things, rather than bad ones. Again, one might answer that this is for reasons of selective adaptiveness, but then the question arises as to why we should encourage selective adaptiveness, rather than its opposite, to which there can be no non-normative answer that does not simply beg another normative question.

    Philosophical inquiry begins with common practice and speech. This may be something to which you would like to object, but there are any number of quite powerful arguments as to why there really is no choice, but to start from—and to some extent, return to—that place. Some of the reasons are Wittgensteinian and have to do with the inherently public and social nature of language (including the language in which we engage in scientific inquiry), while others are of the sort that Stanley Rosen articulates in his 1996 Bradley Lecture, “Philosophy and Ordinary Experience,” of which I will quote only a small part (Brackets are mine):

    “My thesis is not simply that there is an ordinary language, reflective of the common stratum of human nature and accessible under certain historical circumstances without itself deteriorating into a historical artifact. I also claim that this ordinary language is retained as the basis or foundation of all technical dialects [Including scientific dialects]. It is this basis that serves as the fundamental paradigm for the plausibility or implausibility of theoretical discourse…It is not satisfactory to evaluate philosophical [or scientific] doctrines on purely technical or formal grounds because these grounds cannot establish their own validity or authority.” (reprinted in “Metaphysics in Ordinary Language, Yale University Press, 1999)

    So, yes, straightforward but not a solution. As have been all your attempts to dismiss the philosophical, folk-psychological, and normative dimensions of human discourse and inquiry.

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  20. Coel wrote:

    “This is a good example of philosophy and science seeing it differently. In science “truth” is simply descriptive, the word simply denotes a correspondence with reality.”

    ——————

    It is also a good example of where the scientific understanding of a concept is crude and unreflective, in comparison with its philosophical counterpart. If you were at all familiar with the literature on truth, you’d know that the “correspondence theory” is one among many and is beset with what many believe to be crippling and irreparable problems. This is part of the reason why so many who work in the area have become deflationists of one stripe or another.

    To speak unproblematically, as you do, about the idea of truth as correspondence reveals just how unfamiliar you are with the scholarship on the subject of truth. It might be advisable to familiarize yourself with it, if you want to continue speaking out publicly on the subject.

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  21. >>Empirical” doesn’t just mean “pertaining to objective reality.” It specifically refers to the role of the senses/observation/experiment in justifying a claim.<<

    Experiments or observation can apply to mathematics too (thought experiments, thought observation). Physical senses may not be useful in justifying some claims of mathematicians, especially claims about facts that are not part of our physical world, but such facts may be justified by thought, with rigor and certainty that may be even greater than that of physical senses. I think the basic goal of science is to gain knowledge about objective reality, with or without physical senses. It does seem more useful to gain knowledge about physical world but who knows if knowledge about a larger objective reality doesn't prove useful in the future?

    Regarding set theory, I meant to say that IF no contradiction is implied by objects forming a set THEN it is self-evident that they form a set, simply by virtue of their existence. I just wanted to point out that the set-theoretic universe is based on this self-evident principle, with its boundaries naturally determined by the principle of non-contradiction. Finding a contradiction may actually be a difficult epistemic process and it may even be impossible as per Godel's second incompleteness theorem.

    Continuum hypothesis: is it possible that both ZFC with CH and ZFC without CH exist but these two internally consistent systems don't form a set? Perhaps similar to a system of all sets which cannot be a set because it would lead to a contradiction (Russell's Paradox)?

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  22. Hi Aravis,

    Truth and justification are treated as normative, by everyone, in their daily conduct, as well as in ordinary discourse and speech. We encourage people to believe true things, not false ones.

    Agreed. (And note that all of that is descriptive, not normative.)

    Again, one might answer that this is for reasons of selective adaptiveness, …

    Yes, I think it is (the brain evolved as a decision-making tool; fairly obviously it will in general make better decisions if it knows more of the truth; and therefore it will be adaptive to seek and value truth). Again, that statement is descriptive.

    … but then the question arises as to why we should encourage selective adaptiveness,…

    Here you have switched to a normative question, why we “should” encourage adaptiveness. You say:

    there can be no non-normative answer that does not simply beg another normative question.

    Yes, there is such an answer. It is this: there is no reason why you “should” value and encourage adaptiveness. There is no leap to normativity.

    What’s wrong with stopping at the description? People *do* value truth, because evolution *has* programmed them to value it. People *do* act on things they value. This *does* explain why people feel and act as they do.

    Everything is now explained. We don’t need any normative aspect to truth. If people value it and seek it then that’s up to them. But there is no reason why they “should” value it. Why would there be?

    Philosophical inquiry begins with common practice and speech. This may be something to which you would like to object …

    No, actually, I’m entirely comfortable with that starting point. But, we should not accept human intuition at face value. The fact that a human intuits something as normative does not mean that it is. There may be only description. (Relevant also to moral realism).

    So, yes, straightforward but not a solution.

    It seems a straightforward solution to me. What aspect of the topic remains unexplained?

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  23. The quote–and larger article from Rosen–answers your question. As does my point referencing Wittgenstein. I am afraid you will have to investigate these things yourself, if you want more detail.

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  24. Hi Aravis,

    Again, I beseech you, try to resist the impulse to turn everything into a discussion about your superior familiarity with the literature. I agree with the points you are making but it’s really offputting that you expect everyone to be as familiar with scholarship as you are, especially when the points they are making are actually still competitive (as is the correspondence theory of truth, which I would personally endorse).

    Since Coel’s interpretation of truth is defensible, the fact that he does not acknowledge other interpretations or the arguments against his interpretation is not that much of a problem because truth is not what he is particularly interested in debating.

    I think it would have been more productive to just point out that this interpretation of the truth is not without plausible objections. There is no real need to bring Coel’s familiarity with the scholarship into it. The problem is not so much with the points you are making but with your tone.

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  25. Sorry you feel that way, but it is absolutely relevant to the discussion of Scientism.

    Part of the problem with Scientism is that it denies the requirement for expertise in other disciplines. Scientismists like Krauss and Harris go out in public and opine on things they know nothing about–like Ethics–because they think that to be knowledgeable in science is sufficient to discuss every and any topic. In my view, Coel does much the same thing.

    And no, a naive correspondence theory of truth is *not* defensible. Not today. Not anymore than a naive Newtonian mechanics is a defensible view of physics.

    So, while you may think that my “tone” is off-putting, realize that for many of us, it is off-putting when someone is dismissive of the things that *we* know something about. It would be outrageous if I went on Coel’s blog and began spouting off on astrophysics, which I know nothing about, and persistently ignored and rejected the points being made to me by the astrophysicists in the conversation. Well, the same is true here. The trouble is that Coel doesn’t believe that philosophers or linguists have any special expertise with respect to concepts like justification and truth — he has said so in black and white. And so, what he does here is as equally disrespectful as what I would be doing on his astrophysics board, in the hypothetical scenario I described.

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  26. >>Why not? The number of sets that are generated grows without limit.

    Yes, the *number of sets* generated grows without limit, but none of those sets is itself an infinite set (and you need those for most structures in higher math). In other words, the process you described (as I understood it) generates an infinite collection of *finite* sets. To get an infinite set, you either need to simply assume that one exists, or you need something called a ‘reflection principle.’ This is covered in many set theory books.

    >>Regarding set theory, I meant to say that IF no contradiction is implied by objects forming a set THEN it is self-evident that they form a set, simply by virtue of their existence.

    But I’m trying to tell you that there’s no single answer to whether “a contradiction is implied by objects forming a set.” Whether a contradiction is implied depends on your background assumptions. Take the set S I mentioned earlier. Does the existence of S entail a contradiction? Well, if your only assumptions are ZFC, then no, it doesn’t. But if your background assumptions are ZFC+CH, then yes, it does. For another example, suppose C is some set that requires the axiom of choice. Does the existence of C imply a contradiction? Well, if your only axioms are ZF, then no, it doesn’t. But if your axioms are ZF+AD (the “axiom of determinacy”), then yes, it does. Which set of background assumptions does your principle operate on?

    >>Continuum hypothesis: is it possible that both ZFC with CH and ZFC without CH exist but these two internally consistent systems don’t form a set?

    This is a view that some people take (Joel Hampkins, for example, advocates a ‘multiverse’ of sets).

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  27. DM,
    Again, I beseech you, try to resist the impulse to turn everything into a discussion about your superior familiarity with the literature.

    I think you are going too far here. Aravis is the professional and he makes very good arguments that he backs up with references. That is completely normal in academic work. There is a large body of accumulated literature that has established a baseline of well accepted knowledge. Instead of redoing all that work, one simple refers to it as the foundation for your assertions. This is absolutely normal in academic work and your rebuke is really out of place.

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  28. Hi Aravis,

    To speak unproblematically, as you do, about the idea of truth as correspondence reveals just how unfamiliar you are with the scholarship on the subject of truth.

    Actually, I’m entirely ok with the correspondence theory of truth. That is what science means by truth, and, in the context of my article — that maths is a part of science — it is the scientific definition of truth that matters for the arguments in that article.

    It is also what people mean by “truth” in “their daily conduct, as well as in ordinary discourse and speech”, which, as you have stated, is what philosophy starts with. Thus, correspondence-theory-of-truth truth is what “truth” actually means. If philosophers have some other conception of truth then they are talking about p-truth. Which is fine, but scientists and others are entirely within their rights to talk about “truth” in the scientific sense (however imperfectly humans may be able to attain truth). As I said, I don’t accept that philosophical conceptions of these things trump scientific ones. Anyhow, all of the supposed objections to the correspondence theory of truth can be dealt with adequately from a scientific standpoint.

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  29. Hi Aravis,

    The quote–and larger article from Rosen–answers your question.

    Well the quote certainly doesn’t (I’ve not read the full Rosen article).

    I am afraid you will have to investigate these things yourself, if you want more detail.

    The scientific account as I outlined it works just fine and adequately explains everything that needs explaining. It is also markedly superior to an “explanation” that resorts to unreducible and ineliminable non-explanations.

    It’s really up to you to show that there is something beyond that account and show that there actually is something normative about truth (as opposed to people merely thinking that there is).

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  30. Coel, while I agree with DM that some of Aravis’ (and, arguably, my one) comments show a bit of impatience, and while I keep striving for constructive discourse here at SciSal, you simply can’t say either that the correspondence theory of truth is unproblematic, or that that is what “science” means by it.

    Science doesn’t come equipped with a theory of truth, it simply adopts one by default. As for the problems with the correspondence theory, please check the relevant entries in the SEP. They’ll make it abundantly clear that *of course* correspondence is problematic.

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  31. >>In other words, the process you described (as I understood it) generates an infinite collection of *finite* sets.

    Well, an infinite collection is an infinite set, a set with infinitely many members, no?

    >>But I’m trying to tell you that there’s no single answer to whether “a contradiction is implied by objects forming a set.” Whether a contradiction is implied depends on your background assumptions.

    Ok, but since there doesn’t seem to be an objective reason why prefer a certain axiom or its negation, I think I would go with the set multiverse view: in one universe the axiom holds, in another universe a negation of the axiom holds. (Seems similar to the multiverse hypothesis in physics.) But would this mean that there are no contradictions from the multiverse viewpoint? Could the general/naive set theory that says “any collection of objects is a set”, along with its Russell Paradox and other contradictions, be a complete description of the ultimate multiverse of sets, where any contradiction just indicates a split of worlds of sets?

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  32. “The scientific account as I outlined it works just fine and adequately explains everything that needs explaining.”

    ———

    Not for people who actually work on theories of truth for a living.

    Look, you can think whatever you want about truth. But the people who are paid by universities to do research and teach on the subject will have no interest in it.

    Again, it would be as if I spouted a bunch of things about astrophysics, which I knew nothing about. Sure, I could do it, but no one who mattered would care.

    ———-

    As for the rest, the Rosen piece that I quoted is quite clearly a rebuke to the point you have been pushing throughout, for it speaks directly to the question of the relationship between ordinary and technical speech, their relative priority, and the reason for it. That you don’t see it is entirely your problem.

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  33. Coel wrote:

    “That is what science means by truth.”

    ————-

    Then science has demonstrated its ignorance on the subject.

    ————-

    It is also what people mean by “truth” in “their daily conduct, as well as in ordinary discourse and speech”, which, as you have stated, is what philosophy starts with.

    —————

    They also mean it normatively, which you conveniently drop, because it doesn’t fit with you pre-conceived, uneducated view on the subject.

    Certainly, philosophy—and all technical speech—begins with ordinary language and practice and, to a certain degree, is constrained by it, but that doesn’t mean that it is left bare and un-analyzed.

    ————–

    I don’t accept that philosophical conceptions of these things trump scientific ones.

    —————–

    This would be like me saying, “I don’t accept that astrophysical conceptions trump philosophical ones.” Sure, I could say it, but no one who matters would care. And the same is true of what you “accept” and “don’t accept” with respect to things that you have demonstrated you know nothing about.

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  34. I have to concur with labnut. Aravis has provided ample reasons why he thinks Coel is incorrect (reasons I happen to agree with). To upbraid Aravis in the face of an interlocutor who simply insists with certitude that his view is correct misrepresents matters. Aravis and others (including SciSal) have provided ample reasons why they believe that Coel’s position is problematic. These reasons have not been substantively addressed. That indicates (to me at least) that this intransigence is not so much based on a reasoned consideration of relevant viewpoints but rather rooted in what can only be described as a kind of ideological fundamentalism.

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  35. Guys, I appreciate the passion on both sides, but let’s turn the heat down a notch or two, yes? Cheers.

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  36. Hi Massimo,

    you simply can’t say either that the correspondence theory of truth is unproblematic, or that that is what “science” means by it.

    I do think that truth as in the correspondence theory of truth is the basic definition of what people actually mean by “truth”, certainly in science.

    If people have other related concepts it would be clearer to use a different word for them.

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  37. Hi Aravis,

    The trouble is that Coel doesn’t believe that philosophers or linguists have any special expertise with respect to concepts like justification and truth — he has said so in black and white.

    The trouble is rather that you don’t believe that scientists have any special expertise with respect to concepts like justification and truth. Whereas the people who have the best demonstrated ability to get at truth are scientists. Science does work spectacularly well. It just may be that they know something about truth and how to justify it.

    And what I said about philosophers was not a denial of their expertise in such topics, but rather a denial that their viewpoint and expertise automatically trumps that of scientists.

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  38. Hi Aravis,

    “That is what science means by truth.” Then science has demonstrated its ignorance on the subject.

    No, science is simply defining the concept, and thus defining what it is aiming for.

    Certainly, philosophy—and all technical speech—begins with ordinary language and practice and, to a certain degree, is constrained by it, but that doesn’t mean that it is left bare and un-analyzed.

    Exactly. So when science can *explain* why people *feel* truth to be normative, at the same time as producing a better conception of truth in which it is not normative, then that is what science adopts. Especially so when the alternative is to insist that normativity is present but then claim that it cannot be explained, which is what taking it as unreducible and ineliminable amounts to.

    This would be like me saying, “I don’t accept that astrophysical conceptions trump philosophical ones.”

    Sure, and you’d be very welcome to say that. I would not assert that astrophysical conceptions should “trump” philosophical ones — rather, I’d regard the matter as something to be settled on evidence and reason.

    Thus I’d only assert the astrophysical stance if I could produce strong evidence for it. Can you produce any actual evidence, for example, that there is something normative about truth? (Since people’s *feelings* about it can be explained in a non-normative way, those feelings don’t amount to evidence for normativity, so I suspect that you cannot.)

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  39. Coel,
    Massimo defined the heart of the argument here:

    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

    You replied that science uses both correspondence and coherence theory of truth(theoretical physics is a good example of the latter as you mentioned at the end of your post) and also said:

    Maths can still be a part of science even though the mathematicians themselves don’t concern themselves with the correspondence to the real world.

    Massimo replied:

    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.“.

    It really is that simple. Theoretical physics may be an example of how science uses coherence but it is strictly constrained by empirical information in the sense that it is making testable predictions about the world..

    You replied:

    If the axioms of maths are derived from (and thus constrained by) the empirical world…then maths surely *is* constrained by empirical information

    but that was a contradiction of your earlier statement:

    … mathematicians themselves don’t concern themselves with the correspondence to the real world.

    You can’t have it both ways.

    And this is where your argument fails. The foundations of mathematics ‘may’ be constrained in some limited way by the empirical world but nonetheless it ‘has‘ built a large body of scaffolded knowledge that is not constrained by the empirical world(as you admitted, in your contradiction, ‘don’t concern themselves with correspondence to the real world’, in the quote above).

    And therefore it is not science. Knowledge that has no correspondence with the real world cannot be science, regardless of its foundations.

    A simple example of how we can build scaffolded knowledge on an empirical foundation and not be science:
    I can observe my dogs’ behaviour and report it – correspondence theory of truth (you can call it science if you like). I can invent an elaborate fictional tale about the very same dogs, using the same names and descriptions(your axioms). My story is certainly constrained by the needs to be plausible but the scaffolded tale built on top of that bears little relation to actual empirical reality. It is a pure construction of my mind that cannot be verified empirically. It is true in the sense that it is coherent as dogs do behave the way I described. My story is based on observed truths(names and properties of dogs), derived from the real world, but it is fiction, unverifiable and so cannot be science.

    At the heart of all this argument is your refusal to believe that the human mind can construct scaffolded knowledge that is independent of the scaffold’s foundations. But we evidently can as my simple analogy shows and you admitted as much when you said “… mathematicians themselves don’t concern themselves with the correspondence to the real world.“.

    You can’t have your cake and eat it. Knowledge that has no correspondence with the real world cannot be science without eviscerating the meaning of the word ‘science’.

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  40. Is math “constrained by empirical information”?

    There is likely to be theorems whose proofs are so big that it requires humongous computers to check them. Perhaps “practical” is better than “empirical” here.

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  41. Coel, sorry, but no, it doesn’t work that way. Sure, individual scientist, and laypeople, do operate under some sort of intuitive conception of truth. I really don’t think they have anything as sophisticated as a “theory” of it. “People” don’t have “related concepts.” Philosophers – i.e., professionals who specialize in epistemology – have come up with a number of serious accounts of what truth is. And they have uncovered a number of serious issues with the correspondence theory. It may still very well be that there is no practical alternative to it for everyday and scientific uses, but if you want to seriously talk about it, it really is up to you to read about the topic. Here are the pertinent SEP entries:

    http://plato.stanford.edu/entries/truth/
    http://plato.stanford.edu/entries/logical-truth/
    http://plato.stanford.edu/entries/truth-deflationary/
    http://plato.stanford.edu/entries/truth-coherence/
    http://plato.stanford.edu/entries/truth-pluralist/
    http://plato.stanford.edu/entries/truth-identity/
    http://plato.stanford.edu/entries/truth-correspondence/

    Enjoy…

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