The return of radical empiricism

zby Massimo Pigliucci

“All our knowledge begins with the senses, proceeds then to the understanding, and ends with reason. There is nothing higher than reason.” So wrote Immanuel Kant in his Critique of Pure Reason, one of the most influential philosophy books of all time. Kant is also the philosopher credited for finally overcoming the opposition between empiricism and rationalism in epistemology, as he realized that neither position, by itself, is sufficient to account for human knowledge.

Kant was notoriously awoken from what he termed his “dogmatic slumber” [1] by reading David Hume, who had written in his Enquiry Concerning Human Understanding:

“All the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of fact. Of the first kind are the sciences of Geometry, Algebra, and Arithmetic … [which are] discoverable by the mere operation of thought … Matters of fact, which are the second object of human reason, are not ascertained in the same manner; nor is our evidence of their truth, however great, of a like nature with the foregoing. … If we take in our hand any volume; of divinity or school metaphysics, for instance; let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames: for it can contain nothing but sophistry and illusion.”

The second part of the quote makes it clear that Hume, in turn, was reacting to the philosophical excesses of the Schoolmen, the medieval logicians who attempted to discover truths about the world by sheer power of mental analysis — an approach that, to be fair, goes back at the least to Plato himself, who was himself impressed by the effectiveness of mathematics in arriving at conclusions with certainty, and thought that the task of philosophy was to do likewise when it came to its own spheres of interest.

Why am I reminding you of all this? Because I am now convinced that we are witnessing a resurgence of what I call radical empiricism, the sort of thing that we thought we had left behind once Kant came onto the scene, and which, frankly, not even good ‘ol Hume would have endorsed.

Recently, here at Scientia Salon I published three essays — two by Robert Nola [2] and one by Coel Hellier [3] — that epitomize radical empiricism, more so in Hellier’s than in Nola’s case, I might add. Interestingly, Nola is a philosopher and Hellier a scientist, and indeed it is known by now that “scientism” — which is the attitude that results from radical empiricism — is being championed by a number of scientists (e.g., Lawrence Krauss [4], Neil deGrasse Tyson [5]) and philosophers (James Ladyman and Don Ross [6], Alex Rosenberg [7]).

Clearly, I find myself puzzled and bewildered by this state of affairs. As someone who has practiced science for a quarter century and then has gone back to graduate school to switch to philosophy full time I have a rather unusual background that, I think, makes me appreciate where radical empiricists come from, and yet which also precludes me from buying into their simplistic worldview.

In the remainder of this essay, then, I will try to do the following:

  1. Sketch out what I see are the logical moves attempted by radical empiricists;
  2. Show why they don’t work;
  3. Explain why this is more than an academic debate, and certainly more than “just semantics.”

Radical empiricists’ moves in logical space, and why they don’t work

My, by now, extensive readings of and conversations with radical empiricists have unearthed a number of standard moves they tend to make. I will briefly discuss six of them. Two obvious moves are (i) the use of an over-extensive definition of science and the assertion that other valuable disciplines — particularly (ii) logic and math — are “ultimately based” on empirical facts. Since radical empiricists do not seem to value (except for some degree of forced lip service when challenged) any other kind of inquiry or method of understanding (say, philosophy, literature, or the arts), it then follows that science really is all we should care about. It is as if they collapsed Hume’s already narrow distinction above between relations of ideas and matters of facts, arguing that the former are really a version of the latter anyway.

The concept of science, of course, has changed over time. The term did not actually exist as indicating a particular approach to knowledge of the world until recently [8]. Arguably, Aristotle (but not Plato!) was doing science, and so were some of the pre-Socratic philosophers, particularly the atomists. After the Renaissance, “natural philosophy” began to separate itself from philosophy more broadly construed, and finally a number of individual sciences became independent during the 18th, 19th and 20th centuries (most recently psychology, which was still a branch of philosophy until about the time of William James).

But modern defenders of radical empiricism don’t get to help themselves to the fact that what we understand by science has changed over the centuries, because if they did they might have to concede that, really, historically speaking it’s all philosophy.

Where could we turn for help, then? I’d say the dictionary, to get us started. Dictionaries are funny things. They play both a descriptive and a prescriptive role. They are descriptive of how — at any particular moment — a given culture uses a certain term; that, of course, can and does change over sufficiently long periods of time. But dictionaries are also prescriptive in the sense that, within a reasonably short time frame, they also tell us how we ought to deploy those terms. One doesn’t get to arbitrarily redefine words to suit one’s own ideological position or personal inclinations.

So, what are the dictionary definitions of science, mathematics and logic? Here they are (from my built-in Apple Dictionary):

science, the intellectual and practical activity encompassing the systematic study of the structure and behavior of the physical and natural world through observation and experiment. (Interestingly, the same dictionary also provides this alternative meaning: “knowledge of any kind,” but labels it as archaic.)

mathematics, the abstract study of number, quantity, and space.

logic, reasoning conducted or assessed according to strict principles of validity.

It ought to be clear even from these definitions — which are congruent with the vast majority of the specialized literature on the philosophy of science, of math, and of logic — that mathematics is distinct from but akin to logic, and that both of them are very distinct from (although very useful to) science. Hume was onto something, after all.

As I mentioned, the most common refrain from radical empiricists when faced with the above is that math and logic “ultimately” are rooted in empirical knowledge, a recurring example being that we believe that 1+1=2 because we can see that if we put side by side two objects of the same kind we get a total of two objects of the same kind. Another example is that standard practices in logic, say modus ponens [9] are adopted because they “work” in the real world.

Both responses miss the mark because they subtly but surely change the conversation. The first example tells us at most that human beings began to think about abstract objects prompted by elementary empirical observations. But the question at hand is not how mathematical reasoning originated in the Pleistocene, it is what kind of mental activity is modern mathematics. And much of it has nothing whatsoever to do with empirical groundings of any sort. Yes, math is deployed as a tool in science and in all sorts of other applications, but there are huge swaths of mathematical territory that neither describe anything in the world nor are pursued by mathematicians for any practical reason at all.

As far as logic is concerned, a similar reasoning holds there too. And the example of the utility of modus tollens is another red herring that derails the conversation: the question isn’t whether some principles or methods of logic are useful and therefore employed in other areas of application. Of course they are. But logicians — just like mathematicians — are concerned with the formal structure and internal coherence of their constructs, not with whether they do or do not map onto the real world. Many of those structures do not, in fact, map onto the world. When they do, it is only because the world as it actually is does not contain logical contradictions and mathematical inconsistencies, so math and logic are bound to describe the real world together with countless other hypothetical ones (this is true quite irrespective of the ontological question concerning abstract objects, i.e., regardless of whether one is inclined to be a Platonist or not).

Another common move employed by radical empiricists is to (iii) deny the existence of a priori knowledge. It cannot exist, because otherwise they’d have to admit that science (understood as an essentially empirical enterprise) isn’t the source of all knowledge. The most sophisticated of the new wave of radical empiricists sooner or later will cite W.V.O. Quine’s famous rejection of the difference between analytic (a priori, by reasoning) / synthetic (a posteriori, by observation) truths in his paper, “Two dogmas of empiricism” [10]. But I bet that a good number of them have not actually read it, and even more likely that they are not aware of the criticism it got and of the significant amount of backtracking Quine himself had to do throughout the rest of his career.

You see, Quine made ample room for a priori truths in his “rejection” by acknowledging two things: the special status of mathematics as a type of science because it has applications in science (but see above for why this is irrelevant), and the fact that tautological statements (the famous “bachelors are unmarried men” kind of thing) are indeed examples of analytic truths, but turn out to be “epistemically insignificant” according to Quine’s judgment [11]. Well, that’s his opinion, and given that much of logic and math are built on tautologies, a very debatable opinion at that.

A better example of what Quine was talking about are equations such as F = ma from Newtonian mechanics. He thought that this may look like an analytic truth, specifically a definition (hence tautological) of force. But in fact the equation is only true within a specific empirically-based theory of the natural world, its truth not deriving from mathematical reasoning per se. I have no qualms with that, but acknowledging this is a far cry from saying that there are no a priori truths and no difference between synthetic and analytic statements.

Radical empiricists’ next move is to (iv) point out that science uses the same fundamental tools — observation and reason — that we all deploy in everyday life whenever we want to know anything at all. This is just as true as it is utterly uninteresting. It would be surprising, in fact, if science as a human epistemic activity were to somehow transcend the basic intellectual faculties of our species and operate sui generis (just as it would be equally surprising if there were a philosophical method that was entirely distinct from normal human reasoning). Of course doing math, logic, philosophy, art, literature, navigating the New York City subway system, and plumbing use facts (whenever appropriate) which are analyzed by reason. Nevertheless there are tons of interesting distinctions among all those activities, distinctions that are lost by the quest for what I have come to call “explanatory monism,” the obsession with a one-size-fits-all epistemology. Epistemic pluralism is much more interesting and fecund, not to mention more accurately reflective of actual human practice.

The next move, then, is a partial retreat on the previous one, and goes something like this: (v) there are no sharp distinctions between the mentioned activities, so there is no principled way to distinguish among them. To which I can only reply in two ways: there is no sharp distinction separating a helicopter, a jumbo jet and a Saturn rocket, as they are all flying machines. But if you think there are no interesting differences among them you are sorely mistaken. Also, anyone seriously arguing that philosophy, math, logic and, say, biology, are more or less the same thing has clearly not read a single technical paper in more than one of those disciplines.

There is one more defense of radical empiricism, rooted in a kind of greedy reductionism: (vi) the idea that “ultimately” whatever it is we are interested in (poetry, art, mathematics) is made of physical matter or done by beings made of physical matter, so that it all comes down to neuroscience or, if the radical empiricist is particularly bold, to quantum mechanics.

This, again, is a move predicated on shifting the discourse without apparently realizing that one has done so. The issue isn’t what something is made of (ontology), but rather how we may best proceed in understanding it (epistemology). Epistemologists understand very well that for any particular problem X there is a usually small number of levels of analysis that are most informative and appropriate in order to understand X. These can be located one or two (loosely defined) levels of complexity below or above X itself, but the explanatory returns taper off very quickly after that. Let me give you an example.

Let’s say you want to understand the population dynamics of a species of plants, for instance belonging to an invasive species (this comes straight out of my work as an empirical scientist, as you might have guessed). It is of no use to point out that plants, “ultimately” are made of quarks. A quantum mechanical theory of population dynamics — even if possible in principle — is never going to be developed and it wouldn’t help anyway because it would be far too complicated (and unnecessarily so) for a human to comprehend. Instead, the population biologist looks at population genetics (circa one level of complexity below that of organismal biology) and at ecosystem theory (circa one level of complexity above).

Similarly, it is a good bet that to understand economies one needs to operate at the level of economics as autonomous science, plus at the levels of, say, human sociology and psychology. Neuroscience is not likely to be helpful, because it would be too detailed for the problem at hand, even though of course economies are inventions of the human mind, and of course the human mind is the result of the activity of the brain, and of course the brain is made of neurons and other cell types. If you are not convinced, try to go even further down the hierarchy of complexity. How likely is it that we could develop a useful theory of economies based on molecular biology (after all, the brain is made of molecules!)? What about fundamental chemistry (those molecules are made of atoms!)? And so forth until we get to the single wave function that allegedly represents the entire universe.

So, a crucial reason to maintain distinctions among fields of inquiry — even when acknowledging bridges, cross-pollination, and similarities — is that ultimate reductionism will always be a losing epistemic proposition, even if one agrees with the ontological statement that everything is made of quarks (or strings, or wave functions).

Why bother?

I find all of the above intrinsically interesting as an example of intellectual debate about matters of proper definitions, conceptual understanding of different human epistemic activities and so forth. In other words, as a professional philosopher this kind of discussion represents a worthwhile venture into the philosophy of science and in epistemology. But there are far more practical reasons why the assault of the radical empiricists ought to be resisted.

Two reasons in particular are of concern to me: the damage being done to non-scientific disciplines, and the damage potentially to be suffered by science itself.

For years now the humanities and any non-STEM (Science, Technology, Engineering and Mathematics) fields have been in retreat in colleges throughout the world, especially in the US. This retreat is the result of a number of factors, perhaps foremost among them the increasing importation of business-style models into academia and the resulting conviction that if studying a given discipline doesn’t have an immediate payoff in terms of employment then it is not worth studying. This is a false and perniciously instrumental view of higher (and lower, really) education, which has the potential to undermine people’s ability to develop into cultured human beings capable of reflecting on what they do, how they do it , of appreciating all aspects of life (not just jobs and livelihood), and of making informed decisions as members of a democratic polity.

The aggressiveness of radical empiricists and their dismissal of non-scientific fields exacerbates this problem, and in my mind, therefore contributes to undermining the very fabric of our democracy and to decreasing the quality of our life.

This may sound like “defending the turf,” and in a sense it is. But some turfs are worth defending against an all-encompassing cultural imperialism that risks to flatten the intellectual landscape in the name of Science (notice the capital S). And no, I’m not at all coming at this from the point of view of mystical or theological woo in constant entrenchment against science — as I hope is abundantly clear by the body of my writings.

The second worry may seem specious, but I think it is just as important to appreciate. I think that an over-emphasis on the powers and overall reach of Science will, in the long run, do harm to actual, good science. We are already facing a public that is increasingly unwilling to trust scientific findings (just think of the widespread rejection of the theory of evolution or the notion of climate change, or of the uncritical acceptance of a non existent causal link between autism and vaccines, to mention just a few examples). The more scientists are seen as arrogantly dismissive of any other dimension of human experience the more this distrust will grow and fester. And science, the real science done in countless laboratories and university centers across the globe, is just too precious an achievement of humanity to let it be damaged by an emotional reaction to the loud, radical statements of an overbearing but comparatively small number of highly visible public figures.

Isaac Asimov famously said that “The saddest aspect of life right now is that science gathers knowledge faster than society gathers wisdom.” Indeed, but we don’t get wisdom from science alone.


Massimo Pigliucci is a biologist and philosopher at the City University of New York. His main interests are in the philosophy of science and pseudoscience. He is the editor-in-chief of Scientia Salon, and his latest book (co-edited with Maarten Boudry) is Philosophy of Pseudoscience: Reconsidering the Demarcation Problem (Chicago Press).

[1] As he put it, in Prolegomena to Any Future Metaphysics.

[2] Scientism: ‘Yippee’ or ‘Boo-sucks’? — Part I and Part II, by Robert Nola, Scientia Salon, 18 and 19 August 2014.

[3] Defending scientism: mathematics is a part of science, by Coel Hellier, Scientia Salon, 21 August 2014.

[4] Lawrence Krauss: another physicist with an anti-philosophy complex, by Massimo Pigliucci, Rationally Speaking, 25 April 2012.

[5] Neil deGrasse Tyson and the value of philosophy, by Massimo Pigliucci, Scientia Salon, 12 May 2014.

[6] James Ladyman on Metaphysics, Rationally Speaking podcast, 9 September 2012.

[7] Is science all you need?, by Massimo Pigliucci, The Philosopher’s Magazine, 2nd Quarter of 2012.

[8] The very word “scientist” was coined by philosopher William Whewell in 1833, in response to a challenge issued by poet S.T. Coleridge.

[9] Modus ponens.

[10] Two dogmas of empiricism, by W.V.O. Quine.

[11] Quine, W.V.O. (1991) Two Dogmas in Retrospect. Canadian Journal of Philosophy 21:265-274, see p. 271.


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315 replies

  1. Response to Robin’s two remarks on “tautology”:

    Sorry for the length below. However my request for precision
    is not just a sidelight here, but quite germane to the discussion
    of this post, and absolutely central to the essay where Coel asserts
    mathematics and logic to be fundamentally empirical (or part of science,
    if that has any meaning beyond defining the use of the word “science”).

    It is worth remarking first that the only two responders to me give quite different
    definitions, one by Coel and two apparently mutually contradictory
    ones by Robin. This makes one wonder even more about the value
    of the several discussions disputing which example is tautological and which is not.

    Robin “says” and I respond:

    “It didn’t occur to you that some people have other things to do?
    …..Do I really have to explain to you what I mean by “tautology” when applied to Modus Ponens?”
    People who in great profusion use the word “tautology” in back-and-forth arguments about, for example, whether some particular statement is or is not a “definitional tautology” might be expected to be clear about what that noun phrase means in general (not merely as applied to one particular case like modus ponens—science, and especially logic and math, students expect decent definitions, not one or a few examples (implicitly followed by ‘…you get the point…’) The way you phrase this almost gives away the game…do you perhaps subconsciously use “tautology”‘ ‘when applied to’ (in your words) one proposition differently than when applied to another one?

    “Now your quibble appears to be that we cannot use “tautology” in this sense to apply to natural language..”
    I neither quibbled nor pre- nor pro- scribed, but quite clearly granted that some using it in natural language probably had a fairly clear conception of what they meant, and politely requested that they inform me of that conception. Read again what I wrote, and, if you can, give me the quibble and maybe I’ll realize belatedly my impoliteness.

    “Tautologies don’t just appear in first order logic, they are in classical logic too. A tautology is any expression which resolves to “True” irrespective of the truth values of the atoms (or ions as Kleene uses the term).”
    Your first sentence here appears to be attacking a straw man. It is nothing like anything I disagreed with. I’m not sure what you mean by “classical logic” here, perhaps it is propositional logic, sometimes called sentential logic by philosophers. If so, of course I agree; in fact, any even slightly strong and reputable type of formal or symbolic or mathematical logic from the past century or more will have that standard notion of tautology as you more-or-less defined it in your second sentence just above. I do not recall for sure whether you took part in the dispute as to whether or not “1+1=2” is a tautology, but neither you nor anyone else pointed out that it is rather difficult, to say the least, to resolve that formula down into a composite of two or more atoms. So it is trivially not a tautology in the sense you attempted to articulate. And thus it seems clear that some other, presumably more general, definition of tautology is being used by many in the discussion. Is it a quibble to politely ask what that definition might be?

    ‘… not every tautology can be described as “valid”.’
    The word “valid” is used in more than one way in logic texts. Besides being rather shocking for you to assert this about any of those uses, it is a further instance of bandying about a word in a vague manner. However I must admit that I had used the phrase “logically valid formula” without making it explicit. But I did not use it 34 times in the midst of hammer-and-tong disputations about whether particular examples were or were not instances of “tautology” or of “logically valid formula” . Next paragraph I’ll give two ways (among several equivalent ways) of defining it, since your unspecified use of the word “valid” I’d like to respectfully request you come clean on and specify. And perhaps then provide an example of a tautology which is not valid in your senses of those two words.

    A “logically valid formula” for me and many others means in some 1st order language exactly a formula which is derivable from the empty set of premisses by ‘the’ proof system for 1st order (the latter of course being any proof system to which Godel’s completeness theorem applies—no one would use any other system seriously). And that syntactic definition, but using that theorem which was proved earlier than incompleteness, less famous, but very non-trivial, and was Godel’s doctoral thesis, shows that the more common semantic definition of the phrase specifies exactly the same collection of formulas— the semantic definition is: ‘a formula which is true in every interpretation of the language’. (Note that “true” and “interpretation” are defined entirely precisely by Tarski.) And finally, the example “1+1=2” would be, as I likely claimed in my previous, an example of such a logically valid formula, unpacking (a verb so many here love to say) “2”, which is not an official constant within any usual language of 1st order number theory, as the handy abbreviation for “s1”, recalling that “s” is the function symbol informally referred to as the successor function….. You’re welcome….Now what do you mean by “valid” where you used it above?

    “When applied to natural language it can be the case that we are saying this somewhat informally, but I see no problem with that.”
    I cannot think of any possibility for your “this” here other than meaning the definition you previously had attempted using the word “atom” (or Kleene’s “ion”—his book is excellent on logic, but rather unsatisfactory by now as pedagogy; after all it is about 80 years old!) If that really is your “this”, I think you might have a severe problem again with a major disputation above concerning whether some sentence in natural language is or is not a tautology, the sentence about bachelors. Once again you can’t resolve it into two or more atoms, so it is trivially not a tautology in this usual sense in mathematical logic.

    “When used formally calling something a tautology says that the truth of the expression follows from the definition of the terms.”
    Now it looks like you have made a complete switcheroo from your earlier attempted definition of “tautology”. And that is quite apart from what seems to be a misconception about formal logic, which is strictly speaking the study of certain finite strings of symbols. There are no “definitions” of the sort you are trying to talk about here. You’ve got syntax and semantics all mixed up; similarly a mixup of “observer language and the object language”, to quote your later quote of Kleene, where these days “metalanguage” as opposed to “observer language” is the common terminology.
    Pardon the brief diversion, but this sanitary separation of languages stuff is in my opinion absolutely the most fundamental lesson that students need to be taught about modern logic, one which Graham Priest seems to completely avoid, certainly in his article in the Stanford Encyclopedia. And that’s one of the main reasons why I am so dubious (see another reply above) about the supposed contributions of the paraconsistency crew and the relevance crew, and to some extent even the modal crew. I’d not include provability logic (with divine Godel, and Boolos as prophet) nor what is called dynamic logic in program verification logic in CS. Those two are only superficially modal logic, and are major contributions to human knowledge.

    “Calling something a definitional tautology simply distinguishes it from other uses tautologies, for example where an adjective is redundant because the noun it describes always has that that quality anyway. ”
    Readers who can make any sense at all of that as defining the adjective “definitional” when applied to the noun “tautology” have my humblest admiration, and have my request to explain down at my mundane level what they see in it. I did think that a tautology must be some kind of assertive sentence in natural language, if there it could be made sense of at all. But now it seems to have become a noun phrase i.e. an adjective preceding a noun???


  2. Hi stevenj:

    Thank you for the huge effort put into this, and not just the exquisite proof-reading. A few more paragraphs could produce a book, one which would likely be wildly popular in a coffee room for phil students. It would need to be securely chained to one of those comfy chairs where the sometimes valuable mutual education by these students is carried out.

    As for “I don’t know what is so abstract about knots”, I’d suggest you learn it from the textbook of Colin Adams. That might require first learning some more university level pure mathematics, mathematical name-dropping being a different subject. It often seems that the number of words on some subject is inversely proportional to what the writer knows about the subject. This is especially so in an academic discipline whose domain is said to be anything and everything.

    “…it was news to me that Godel didn’t really prove much of anything…”
    Was it maybe Fox News?


  3. As I always say, those who ignore much of what passes for results of research in present day academic philosophy departments (e.g. do read the Stanford Encyclopedia discussion of “I”, entitled there “Identity”—or rather, don’t), leave themselves the possibility of actually contributing something valuable to human knowledge.

    The logical axiom “x=x” is indispensable in 1st order logic with “=”, but is there really more needing to be said?


  4. pete1187: Perhaps it was pithy but I very much doubt it was well said enough to give any pause to the crusaders against scientism.

    phoffman56: Thank you so much for checking for typos, misspellings and grammatical errors. It is always good to know someone has their priorities and follows up on them.

    After you’ve been so charitable in reading an appallingly long comment, it pains me to point out that “knots” and “knot theory” are not synonyms. This matters in the context of deciding whether mathematics is the study of mathematical objects rather than empirical reality. That’s why I also mentioned logic, thought, according to Boole. Thinking may not be tangible, but it’s not at all clear in what sense it could be deemed mathematical.

    It was the comments that informed me that Godel’s theorem had no consequences for the proof of mathematics to find truth by a coherentist account. But if you want Fox News politics, just stick with Popper and Hayek!


  5. Your paragraph was:

    “In this presentation, the mathematician studies consequences of axioms as other criteria than the crassly empirical ones of science. It is merely a coincidence if this happens to be helpful to the scientist, but mathematics is deductions from axioms. Sometimes, it is said that mathematics is the study of abstract objects. I can see the appeal of such a definition but I don’t know what is so abstract about knots. In a different vein, Boole and de Morgan did logic (or The Laws of Thought) as mathematics. The thing there, in what sense is logic a mathematical object?”

    I’m just not clever enough to realize that this paragraph is not about mathematics, but rather about fishermen avoiding loss of a boat while attending classes on land at the College of Philosophy.

    To learn about your last question, you could try Manin’s book on mathematical logic, where he introduces 1st order with the paragraph:

    “As a means of communication, discovery and codification, no formal language can compete with the mixture of mathematical argot and formulas which is common to every working mathematician. However, because they are so rigidly normalized, formal texts can themselves serve as an object for mathematical investigation. The results of this investigation are themselves theorems of mathematics. They arouse great interest (and strong emotions) because they can be interpreted as theorems about mathematics. But it is precisely the possibility of these and still broader interpretations that determines the general philosophical and human value of mathematical logic.”


  6. You are so right about the proofreading. One “as” should have been “by,” and a “sometimes” should have been “alternatively.”

    Thank you for providing that paragraph from Manin (“unemcumbered by philosophical or normative prejudices…” as Amazon tells us.) When Manin tells us the “formal texts themselves can serve as an object for mathematical investigation..” it is at least clear. Mathematics is the study of whatever is written in mathematical form, as symbols, equations, whatnot, like graphs and non-pictorial figures. Good to know. Thus, knot theory doesn’t study knots, it studies the formal texts with symbols etc. and it’s just called knot theory for whatever irrelevant reason.

    The investigations themselves can be “interpreted as theorems about mathematics,” which means that there’s nothing applied or crudely empirical about mathematical logic. It is instead “the still broader interpretations that determines the philosophical and human value of mathematical logic.” This is very gratifying to read. I had written “…in order to know what your deductive knowledge means about anything other than the subject, you have to know more, from experience.” Of course I prefer Manin’s way of expressing the need for still broader interpretations as both more general in scope and more specific as to the nature.

    As to you response to my observations that knots and knot theory are not the same thing, and that I was talking about mathematics presented as the study of mathematical objects? “I’m just not clever enough to realize that this paragraph is not about mathematics, but rather about fishermen avoiding loss of a boat while attending classes on land at the College of Philosophy.” You are completely unfair to yourself. This is a perfect example of good philosophy directing the public intellectual in his work. Your post exemplifies Scientia Salon at its finest.


  7. Hi labnut,

    So all you are saying is that your intuitions contradict my intuitions.

    Correct! And what DM and I are trying to point out to you is that all that *you* are saying is that your intuitions contradict our intuitions. If my response was not a serious argument, owing to this property, then nor were your comments, owing to the very same property.

    Question-begging has a well defined meaning and it is not what you claim.

    You start with an *intuition* about the truth of the matter, and you attempt to turn that into an argument, but really your argument just reports your prior intuition. (Which can indeed be properly referred to as “begging the question”.)


  8. “..Mathematics is the study of whatever is written in mathematical form..” is a characterization of mathematics which only needs supplementation by adding that
    ‘mathematical form is whatever is written in such manner as to be regarded as mathematics’. Can we now decide whether Coel was correct, using these two superb definitions?

    Perhaps a slip of the finger led to omitting the ending of that Amazon blurb for Manin’s book:
    ‘ “unencumbered by philosophical or normative prejudices” such as those of constructivism or intuitionism. ‘
    Perhaps all for the good, given the singularly barren landscape of accomplishments the latter two ‘isms’, after a century of efforts. It was Turing and von Neumann who invented the computer, not constructivism.


  9. Hi Coel,

    You see, I asked you to explicitly answer a particular question, even made it bold so you would not miss it and you avoided answering it and started saying other stuff that didn’t answer the question. That suggests to me that your misunderstanding of my point is wilful rather than an inability to understand what I am saying. Answer the question.

    Now if you think that it makes a difference to put “provisionally” between asterisks, then so be it. Here are the steps I listed below with asterisks added:

    1. *Provisionally* adopt I and NC and test model and get result A
    2. *Provisionally* drop I and NC and test model and get result B
    3. *Provisionally* adopt I and NC and test result A against B
    4. *Provisionally* drop I and NC and test result A against B

    So the best, overall, considering all outcomes of 3 and 4, is that I+NC+A works best. It works better than B and better than (~I)+(~NC)+B.

    Now you have added another step, held up another finger to the dealer so to speak.

    Again I ask you to explicitly answer the question – for this step are you *provisionally* adopting I and NC?. If not then clearly you cannot make that comparison from all the outcomes. so we have steps 5 and 6:

    5. *Provisionally* adopt I and NC and consider all outcomes from steps 3 and 4 and see which works the best.
    6. *Provisionally* drop I and NC and consider all outcomes from steps 3 and 4 and see which works the best.

    So you will now have found that, overall I+NC+A works the best, just so long as you first assume I and NC.

    Care to hold another finger up to the dealer?


  10. I mean, seriously, what is your point?

    The logical axiom “x=x” is indispensable in 1st order logic with “=”, but is there really more needing to be said?

    Also indispensable in propositional logic, but so what? How does that relate to anything that anybody has said here?


  11. “Can we now decide whether Coel was correct, using these two superb definitions?” I’m afraid I didn’t understand this is about whether Coel is right.

    Anyhow, I had hoped we were done, but since you insist on asking, I can’t. The empirical element in mathematics, such as measurement, is why it is not unreasonable for mathematics to be effective in science. The more abstract parts for which no reasonable empirical element other than the mathematician and whatever the math is written or drawn on seem to provide valid deductions but it’s hard to say what they mean. If that were so straightforward, we could conclude superstring theory is not even scientific speculation but fact. In many respects mathematics is a cultural invention like writing or money. It’s not a discovery of something that exists elsewhere, inasmuch as no one can quite give directions to where ever mathematics would exist independently.

    As to constructivism or intuiionism, I’m not very sympthetic to efforts to define the scientific method, so I have no idea why I should be concerned about defining the proper mathematical method. I don’t even think that it has to be axiomatized, not necessarily even proof-driven. I don’t think you can get much more anarchic, non-normative or philosophically unprejudiced than that.

    Those two superb definitions are only usable by such talented people as yourself.


  12. That’s fine—we can be done. I had thought most of this exchange was more jocular than serious, and am happy to let your middle two paragraphs just above be your final word uncontested by me.

    But just to explain a couple of my earlier remarks you mentioned just above:

    The word “superb” was intended as ironic, considering that my additional ‘definition’ made the two ‘definitions’ obviously circular. I thought maybe that would be a humorous way of saying that math articulated as being something expressed in “math form” wasn’t really saying much. And perhaps you were joking in writing that. Hard for me to tell.

    The Amazon blurb re Manin’s book which I suggested needed to be not quite so short just seemed, without that addition about intuitionism etc., to give the impression that Manin might be some kind of lightweight who couldn’t be bothered with general questions about what he’s doing. You might have a quick look at the wiki entry on him just to see that he is a scientist of considerable depth and breadth. And the blurb he likely didn’t write, and quite possibly didn’t even read, if my sense of this level of scholar is at all accurate. I have known a few not far below that level, but I’m far from being deep and creative like that.

    Oh, and just a brief suggestion related to your paragraphs: I suspect that the sense in which you are thinking of the unreasonable effectiveness of mathematics might be fairly different from the sense in the physicist Wigner’s famous old paper with that title. He is mainly ‘bemused’ by how often a purely math-curiosity-driven theory (e.g. Riemannian geometry) ends up later being just what is needed in some deep physical theory (e.g. general relativity).


  13. Coel,
    phew, you have completely lost me with all this circular reasoning. My head reels. It is time to stop and take a deep breath. I usually do that by going for a long trail run, or at the very least a good walk. Which brings me to my next book recommendation for you:

    A Philosophy of Walking by Frédéric Gros

    Frédéric Gros is both a philosopher and a walker.

    It is a remarkable thing, people have reported again and again that the act of walking is beneficial to the thinking process. Why that should be so would make an interesting discussion.


  14. Here is a really interesting book recommendation that I received from Thomas Jones. I have just got the book, so I am looking forward to some good reading over the weekend:

    The Island of Knowledge. The limits of science and the search for meaning, by Marcelo Gleiser(

    Here is a review of the book:

    Science as Salvation?
    Marcelo Gleiser wants to heal the rift between humanists and scientists by deflating scientific dreams of establishing final truths.

    Whether or not scientists are from Mars and humanists from Venus, the “two cultures” debate about the arts and sciences has never been down to earth. For decades we’ve endured schematic sparring between straw men: humanists claim that scientists are reductive, scientists find humanists reactionary. (A recent bout between the cognitive scientist Steven Pinker and the literary critic Leon Wieseltier in the pages of The New Republic ran true to form.) Marcelo Gleiser, a physicist with strong ties to the humanities, is alarmed by the hubristic stance of his discipline and the backlash it is liable to provoke. He has written The Island of Knowledge as “a much needed self-analysis in a time when scientific speculation and arrogance are rampant…. I am attempting to protect science from attacks on its intellectual integrity.

    Coel, I am going to recommend you read the book and hear the reply that you have no time to read my excellent book recommendations!

    Massimo, this raises an interesting idea for Scientia Salon, that you or interested readers, publish book reviews on Scientia Salon. They could be full length reviews or short reviews. Mind you, I am not volunteering right now, unless of course, you want a review of Alice Through the Looking-Glass!


  15. Hi Robin,

    for this step are you *provisionally* adopting I and NC?

    Yes, I am provisionally adopting I and NC. If I do that it works; if I don’t do that nothing works. Thus doing so gives me more explanatory and predictive power about the reality we experience (which is all that science claims).

    Or, if we really get down to basics, animals with brains that started working along the lines of I and NC left more descendents. That statement still says that I and NC derive from the real world.


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