I have written a piece that will be published shortly on another site; it’s largely about whether academic disciplines, including the arts, can produce “propositional truths”, that is, declarative statements about the world that are deemed “true” because they give an accurate description of something in the world or universe. Examples are “Jerry has five fingers on each hand”, “Sheila plays the violin in an orchestra,” or “humans and other apes shared a common ancestor.” The reason I was concerned with propositional truths is that it’s often said that the search, production, preservation, and promulgation of such truths is the primary purpose of universities. Is it? Read my piece, which will be out next week, to see. I’ll post a link when it’s up.
I won’t give my thesis here about truth and the various academic disciplines, as that’s in the other article, but in my piece I omitted two areas: mathematics and philosophy. That’s because there’s a big controversy about whether these disciplines do produce propositional truths or, alternatively (and in my view), give only the logical consequences of assumptions that are assumed to be true.
For example, a “truth” of mathematics is that 16 divided by 2 equals eight. More complex is the Pythagorean theorem: in a right triangle, the square of the length of the hypotenuse is the sum of the squares of the other two sides. This is “true”, but only in Euclidean geometry. It is not true if you’re looking at triangles on a curved surface. The “truth” is seen only within a system of certain assumptions: geometry that follows Euclid’s axioms, including being planar. All mathematical “truths” are of this type.
What about philosophy? Truths in that field are things that follow logically. Here is a famous one:
All men are mortal
Socrates is a man;
Therefore Socrates is mortal.
Well, yes, that’s true, but it’s true not just because of logic, but because empirical observations for the first two statements show they are propositional truths! If they weren’t true, the third “truth” (which was tested and verified via hemlock) would be meaningless.
Here’s another of a similar nature that came from a friend:
“All As are B; x is an A; therefore x is B—doesn’t depend on the content of A and B: it’s a *logical truth*.”
Again, the statement is indeed a logical truth, but not a propositional truth because it cannot be tested to see if it’s true or false. Nor, without specifying exactly what A and B is, can the empirical truth of this statement be judged. I claim that all philosophical “truths”—logical truths without empirical input—are of this type.
When I told my friend this, I got the reply, “This is analytic philosophy. The people who do it work in philosophy departments and call themselves philosophers: and most philosophy BA and PhD programs require a lot of it. I’m sure any of our competent philosophers would be happy to supply hundreds of propositional truths that are philosophical.” The friend clearly disagreed with my claim that philosophy can’t by itself produce propositional truths. Insofar as philosophy is an important area of academia, then, I am not sure that it’s discipline engaged in producing or preserving truth.
Two caveats are in order. First, this is not meant to demean philosophy or argue that it doesn’t belong in a liberal education. It certainly does! Philosophy, like mathematics, are tools for finding truths, and indispensable tools. Philosophical training helps you think more clearly Unlike many scientists, I see philosophy as a crucial component of science, one that is used every day. Hypotheses that follow logically from observations, as in making predictions from observations (e.g., Chargaff’s observation, before the structure of DNA was elucidated, that in organisms that amount of A equals the amount of T, and the amount of G equals the amount of C), are somewhat philosophical, and certainly logical. Dan Dennett is a good example of how one can learn (and teach others) to think more clearly about science with a background in philosophy.
Second, I do not feel strongly about what I said above. I am willing to be convinced that mathematics (but not necessarily philosophy) gives us propositional truths. There is, for example, a school of philosophers who accept “mathematical realism,” defined this way in Routledge’s Encyclopedia of Philosophy:
Mathematical realism is the view that the truths of mathematics are objective, which is to say that they are true independently of any human activities, beliefs or capacities. As the realist sees it, mathematics is the study of a body of necessary and unchanging facts, which it is the mathematician’s task to discover, not to create. These form the subject matter of mathematical discourse: a mathematical statement is true just in case it accurately describes the mathematical facts.
An important form of mathematical realism is mathematical Platonism, the view that mathematics is about a collection of independently existing mathematical objects. Platonism is to be distinguished from the more general thesis of realism, since the objectivity of mathematical truth does not, at least not obviously, require the existence of distinctively mathematical objects.
A corollary of this is my own claim (which is mine) that although the objects and “truths” of mathematics and philosophy are inapplicable to all species outside of our own, as only Homo sapiens can grasp, discover, and use them. The earth spins for all creatures and plants upon it, but the integers and prime numbers are “real” only for us. (Do not lecture me that crows can count!).
I have read some of this controversy about mathematics, but it rapidly becomes abstruse and tedious, and so I’m proffering the view of a biologist, not a professional philosopher. I am more open to the idea of mathematics producing truths than philosophy, simply because, as one reader once commented, “You can’t find out what’s true by sitting in an armchair and thinking.”
So it’s clear I’m soliciting readers’ views here to help clarify my own thinking. Comment away!
Some critics of science claim it too is based on faith, like religion. The supposed faith is that the world is constant and predictable. Alternatively, one can think of these as hypotheses tested by science. Something like:
If the world was not constant and predictable, science would fail (not succeed).
Science does not fail (it succeeds).
Therefore, the world is constant and predictable.
Not sure where this fits in the notion of propositional truth. Indeed, one could say all scientific propositional truths “depend on” logic (philosophy?).
If theory X, observation Y
Not observation Y
Therefore, not theory X
Evolutionary epistemology argues science works this way. Rigorous challenging of propositions leads to (more) “truthful” ones that survive the filters.
I’ve heard that Beethoven’s (who was not 100% deaf ) pitch sense drifted about 1/2 note flat by the time of his last compositions.
Not sure what that means, except that a simple “constant” like perceived pitch could depend on age – the individual being the entity which changes – so a tuning fork at A440 will be flat – to them – after however many years.
But the world is not constant (global warming), and quantum mechanics says that it it is not predictable, at least as far as we know. So one premise is simply wrong.
I thought about including qualifiers like “reasonably” or “somewhat” in front of constant and predictable, but decide to keep it simple, overly so perhaps.
How can any scientific equation be correct or true if the world as we know it does not remain sufficiently constant for the equation to continue being valid? And I’m not sure that quantum mechanics qualifies what to predict when I release an object from my hand.
Saying “quantum mechanics says that it [the world] it is not predictable” is a bit of an exaggeration. QM says rather that on the microscopic scale, predictions are in the form of probabilities. Ballentine’s interpretation, saying that QM applies to ensembles of objects and therefore average values of observable physical quantities, gets around the problem. And to be picky, the equations are quite deterministic; it’s the interpretation (the Born rule) that throws in probability, interpreting the wave function (which is pretty much gone in quantum field theory and the Standard Model) as a probability amplitude.
Critics who claim science is based on faith, like religion, err by failing to define religious faith. Religious faith (dogmatism) is the acceptance of a proposition as true even when the evidence is greatly against it being true. To claim science is dogmatism is not a valid criticism.
When we say “the world” is constant we mean only that the universe (not our earth) everywhere is predictable under the temporarily and spatially constant laws of physics to the extent we can know them, limited at very small scales by quantum uncertainty and fuzziness. Changes of state (like the heat content of the atmosphere, or oxidative phosphorylation) are predictable from inputs. Indeed, understanding how things change and verifying predictions about future changes according to a theory of change, is one of the strongest activities of science.
Jerry, I would largely go along with you here, except when you say: “… the integers and prime numbers are “real” only for us”.
As i see it, mathematics, logic and philosophy attempt to describe the universe, in the same way that “laws of physics” are descriptions of the universe. Axioms of logic or mathematics are thus fundamental statements of a model that describes the behaviour of the universe, in the same way that Newton’s laws are fundamental statements of a model (“Newtonian mechanics”) that describes the behaviour of the universe.
Whether a mathematical or logic statement is “true” (in the sense of being a valid world model) can only be determined empirically (where we would be asking whether those axioms do indeed model the world).
But it then follows that, since integers are adopted as being an empirically valid real-world model, that alien scientists from the planet Zog would also arrive at integers and prime numbers (and also Newton’s laws). So they are not peculiar to us.
[If anyone wants to counter by pointing to things like Banach-Tarski and the Axiom of Choice, and asking whether they are real-world true, then my reply would be that these have the same status as hypothetical alternative “laws of physics” that a theoretical physicist might scheme up to explore the conceptual space of possible real-world models.]
● The axiom of choice is proven to be neither provably true nor provably false [insert the usual technical caveats]. It is undecidable. This allows the very safe empirical prediction that no mathematician, biological or inorganic, now or in the future, however clever and persistent, here or in any alternative universe, will manage to prove or falsify it.
● The Zogite Imperium (Hail Zog!), because of their particular biological and cultural evolution, might never have paid any attention to yucky discrete things like integers. There, everything really is a spectrum 🙂.
● A mathematical model of something empirical can be more-or-less accurate, useful, blasphemous, or whatever; these are contingent empirical truths. Whether or not it is internally consistent is a universal logical truth (which might be undecidable!).
But the elements on Zog must be the same as everywhere else in the universe, albeit very likely in different proportions. Zog chemists must have noticed that there was a constant integral relationship in the way elements combined. Indeed if an experiment shows a relationship that doesn’t reduce to simple constant ratios of integers in the formation of, say, rust, it’s a clue to there being two species of one or both elements in the test sample with different oxidation states. In FeO and Fe2O3, the combining ratio of oxygen to iron will depend on the proportion of Fe++ and Fe+++ in that particular sample.
If the Zoggians hadn’t invented integers (even for procreation and counting sheep, both of which could be communal activities not needing counting) they would have had to have invented them to understand chemistry.
Fie, you chemo-supremacist. The Zoggians are plasma beings. Hail Zog!
From a purely logical point of view, the axiom of choice has the same status in Zermeko-Frankael set theory as Euclid’s parallel postulate has in geometry. Euclidean geometry with the parallel postulate is logically consistent if and only is Euclidean geometry with the negation of the parallel postulate is logically consistent. In the same way, set theory with AOC is consistent if and only if set theory with the negation of AOC is consistent.
But psychologically, at this point in history, mathematicians feel differently about the two situations. Nobody frets anymore (not for over a century) about the parallel postulate. We just say that Euclidean geometry with the PP describes the points and lines of a flat plane while nonlinear geometry (without the PP) describes the points and geodesics on a surface of constant curvature.
But the status of the axiom of choice feels different, at least to me, in a philosophical way though not in a purely logical way. I can’t help feeling that there is some kind of “right” concept of sets to which our axiomatic theory can more accurately or less accurately describe. Perhaps I am thinking of the “perfect” class of sets that lives in eternal perfection in Plato’s realm of forms.
There is a long list of propositions that are logically equivalent to the axiom of choice (well ordering theorem, Zorn’s lemma, Hausdorff maximal principle, the high chain principle, the trichotomy principle and many others).
To me, it is the trichotomy principle that seems most obvious “true”. For sets X and Y we define X < or = Y if there exists an injective map X to Y. The Schroeder-Bernstein theorem gives the reassuring result that if X < or = Y and Y < or = X then X is isomorphic to Y. The trichotomy principle states that any two sets X and Y are comparable (meaning that X < or = Y or Y < or = X or both). The axiom of choice is equivalent to the trichotomy principle.
Intuitively I feel that the most natural conception of sets would require any two sets to be comparable. The trichotomy principle intuitively seems to me to be more clearly “true” for the sensible conception of sets than the axiom of choice itself or any other of the dozens of equivalent propositions.
● “The axiom of choice is clearly false and Zorn’s lemma is clearly true”
(a comment on intuitions, commonly attributed to the logician Leon Henkin)
● The axiom of choice, well-ordering, trichotomy etc. are intuitive and readily proven for the smallest infinite sets. But bigger ones are unimaginably huge, and IMO more counterintuitive than quantum mechanics. What you think you see is often not what you get. There are pragmatic arguments in favour of accepting the AoC based on its usefulness, power, etc.; but that’s not the same thing as truth (sort of like the arguments that religion is comforting).
The version I heard was “The axiom of choice is clearly true, the well ordering theorem clearly false, and as for Zorn’s lemma, who can say?” To me anyways, trichotomy seems most obviously “true”. I also find Hausdorff maximal principle more intuitively true than AOC.
But what do we mean by “mathematical truth”? From the purely logical point of view, set theorys with or without AOC are equally good. My preference for AOC true rather than false is not a logical matter. It is a matter of which axiom system more aptly characterizes one’s intuitive ideas about sets.
Since Banach-Tarski is clearly intuitively false, and since BT is true if AoC is true, wouldn’t you say that AoC is thus clearly intuitively false (at least for the sorts of pathological sets that lead to BT)?
The axiom of choice implies there are unmeasurable subsets of the real line, and other things that are “interesting” but unfortunate. Makes me think we should accept its negation, which we can since it’s logically independent of the other ZF axioms. But most mathematicians like the axiom of choice. It’s interesting that we have that choice!
Well, if you negate the axiom of choice then you can measure every set of reals and you can’t do much of anything else! I am less clear whether I intuitively believe the continuum hypothesis.
Love this line of thought! Looking forward to that piece – meanwhile :
PCC(E) : “A corollary of this is my own claim (which is mine) that although the objects and “truths” of mathematics and philosophy are inapplicable to all species outside of our own, as only Homo sapiens can grasp, discover, and use them. ”
Pretty sure that fits into Common Sense Realism.
My theory which is mine is that everything – necessarily – reduces to Scottish Enlightenment / Common Sense Realism.
[ break ]
Question : Are imaginary entities useful, or to be trusted? How about the square root of a negative number?
Answer : Yes. Very.
IMHO I personally can sense, at some point, a “taking things too seriously” energy, in these sorts of meta-analyses. Consider :
“All models are wrong. Some are useful.” (George E. P. Box, 1919-2013 )
And :
“Out of the crooked timber of humanity, no straight thing was ever wrought.”
-Immanuel Kant
Translated from Idea for a General History with a Cosmopolitan Purpose, ca. 1784
Proposition 6.
And perhaps a corollary like Einstein’s famous one would be useful :
[any topic ] should be taken as seriously as possible, but not any more than that.
Awesome topic – let’s go!
Calling the square root of a negative number imaginary is misleading, but that’s the terminology we’re stuck with. The square root of -1 is not a number. It is a rotation operator that rotates 90 degrees counterclockwise. Thus if a vector is in the first quadrant of a Cartesian graph, multiplying by the square root of -1 places the vector in the second quadrant. Suggested reading: An Imaginary Tale (The Story Of The Square Root Of -1) by Paul J. Nahin.
On the other hand, aren’t all words and numbers imaginary entities in the sense of being abstract? They represent ideas. They are not the ideas themselves. And yet they are useful.
Simulacra and Simulation
Jean Baudrillard
1981 Éditions Galilée (French)
1983 Semiotext(e) (English)
1981
Thanks for this post! I agree that mathematical models “give only the logical consequences of assumptions that are assumed to be true.” Mathematical ecologists, who I’ve spent a career admiring, collaborating with, and also arguing with, have a habit of stating conclusions of models as if they were empirical discoveries. They aren’t! Dissect any model and you’ll find the conclusions depend on premises that are “true by convention” or “reasonable assumptions” as opposed to having factual support. At their best, models stimulate new testable ideas; at worst they proliferate into ever-more-elaborate houses of cards.
vide supra
😁
But aren’t the meanings of words themselves typically ‘true by convention’? Eg, ‘cat’ means cat because we share conventions about meaning. If so, then it’s not clear that ‘true by convention’ should be a serious limit on what we usually mean by ‘having factual support’.
As a general point for the wider discussion here, it’s worth noting that a logician’s models are not usually restricted to ‘the way things are’ — they also describe ‘the way things could be’. So they are not as concerned with determining the (empirical) facts of the matter as they are concerned with mapping all (consistent) possible ways things could be — maybe call them ‘situations’ — where one such situation corresponds to the way things actually are.
How do scientists make the possible situations useful? An easy example is that sometimes we don’t know which situation we are in — eg, a blind man who needs to know whether he’s in a situation where there’s an obstacle, say a table, blocking his path. When designing the robot to guide him, we programme it to eliminate possibilities, to hone in on what is actual because that is what the blind man needs. He doesn’t need all the possibilities, just the actual one.
What I meant by ‘true by convention’ in the ecological modeling context is, for example, assuming that the reproductive rate of an organism falls off with increasing population density according to a certain curve. That is an example of a standard, off-the-shelf model ingredient that is considered acceptable to use simply because it’s commonly used. It might be empirically wrong, though, in any given case, and this might make a model’s conclusions wrong. Modelers seldom give much attention to what’s called structural uncertainty — the potential for error caused by wrong structural assumptions.
“All men are mortal. Socrates is a man. Therefore Socrates is mortal. Well, yes, that’s true, but it’s true not just because of logic, but because of empirical observation for the first two statements are propositional truths! If they weren’t true, the third “truth” (which was tested and verified via hemlock) would be meaningless.”
In this case, then, analytical logic has produced a “new” propositional truth, given that the two antecedents were shown to be true empirically. Logical analyses are a way of producing (or revealing) empirical truths based on other empirical truths. Math is the same. I suppose we could argue that these deductions are not new propositions but are implicit in the two empirically confirmed propositions. But I think it is more complicated than that.
Both math and logic depend on assumptions, as you said. But the assumptions underlying logic are themselves empirical choices, chosen because they produce empirically verifiable truths when the premises are true. In this sense logic is empirical.
Several people have eloquently argued this point that logic is empirical. Hilary Putnam famously presented this argument in the 1980s, even raising the point that perhaps we are not actually using the “correct” logic.
https://link.springer.com/chapter/10.1007/978-94-010-3381-7_5
Math is similar.
When Einstein sat in his chair and did his thought experiments, did he “discover” empirical truths about the universe? Many people think that he did. His choice of non-Euclidean geometry was an empirical matter, and if it had not existed, he would have had to invent it.
The dependence of math on assumptions is actually not so different from the dependence of empirical “facts” on the assumptions of science. There is a large body of work on the dependence of so-called “facts” on theory. I always argued against it, but I do think there is some truth to it.
Edit: I see that Coel (Comment 2) made some similar points while I was writing this.
Exactly. I would add that we don’t have to conclude that all familiar mathematical propositions are true. But for the ones that figure into successful scientific theories, it’s hard to see how the laws can be true if the math is not.
In mathematics, theorems follow, by logically sound steps, from axioms. No set of axioms is objectively true, but they can be extremely useful, e.g., the axioms of Euclidian geometry and Zermelo-Fraenkel set theory. So nothing in mathematics is objectively true without the caveat that it follows from certain assumptions.
I think this same thing is true in philosophical areas like moral theory, or the existence of free will. For example, Sam Harris treats moral theory as an axiomatized system. If you accept his basic premises (e.g., a world where everybody is in a constant state of torture is bad) then (assuming his logical arguments are correct) his conclusions are valid. He has not discovered objective moral truths. Likewise, if materialism is correct then there is no libertarian free will.
Science is kind of like this too. Relativity is a very top-down axiomatic theory. And as Einstein said, if there had been experimental results violating it, he would have rejected the experiment rather than the theory. And it might be better to invent some additional new forces or effects to explain contrary results and preserve the axioms, since they have been so successful.
We see something like this going on in physics today. Dark matter was invented to preserve general relativity. A few physicists, however, prefer to reject general relativity. Regardless of which approach wins, the controversy reveals a lot about the nature of science.
Einstein might not have rejected relativity if it didn’t make good predictions or was verified by experiments, but sure as shooting scientists in general would have. Why haven’t they all bought string theory? No way to test it but it makes logical sense?
No, they don’t necessarily reject such theories. See my comment about dark matter. There is a difficult-to-explain notion of “beauty”, perhaps better called “power”, of a theory, that rightly counts a lot in our judgement of what is true. Relativity is really beautiful, in the sense that a tiny set of simple postulates lets us predict most of the universe in great detail, to many decimal points. The appearance of contradictory evidence won’t always make us discard this theory immediately. People will just try to make an ad hoc fix, like dark matter. People will often only reject the central theory when another equally compact and beautiful and successful alternative theory is discovered.
While premature abandonment of a theory is unwise, at some point a theory must be rejected if experiments fail to confirm it, even if one is as confident as Einstein. He was, of course, forced to forsake his doctrine of “God doesn’t play dice with the universe” by the facts of Quantum Mechanics.
It’s tricky; you have to judge a whole constellation of intertwined theories. One or two “disproving” experiments are not enough to kill a beautiful successful theory (and this is a good thing, not a bug). This is the main thesis of Kuhn’s Structure of Scientific Revolutions; the Popperian view of strict disconfirmations does not describe science correctly.
Comment by Greg Mayer
Lou–
What you describe, both here and above, is more akin to Lakatos’ views than Kuhn’s. Your statement above, “People will just try to make an ad hoc fix, like dark matter. People will often only reject the central theory when another equally compact and beautiful and successful alternative theory is discovered.”, seems very Lakatosian; I thought that that’s who you had in mind. He wrote of a central core of a research program, with a belt of surrounding ideas that can be modified without refuting the core.
Lakatos would insist that the ad hoc fix be independently testable; if all you get is one ad hoc after another, your research program is, as Lakatos would say, degenerating.
The classic case of changing the belt of surrounding ideas to save the core is the proposal of an additional planet in our Solar System in order to “save” the core of Newtonian mechanics (which was otherwise so successful). But the ad hoc proposal was independently testable, and Neptune was indeed found when its existence was tested.
GCM
Someone on a photography forum today linked to this very relevant talk by Nobel laureate Murray Gell-Mann about the relative roles of beauty versus experiment in physics.
https://www.ted.com/talks/murray_gell_mann_beauty_truth_and_physics
I was once taught that logic basically boiled down to the elimination of contradictions, and that all philosophy began on the bedrock of A = A — a thing is what it is (this then leads to A =/= NotA, a thing isn’t what it isn’t.)
Is this very very simple foundation a discovery, an intuition, or the discovery of an intuition? If we lacked the ability to understand that A = A — or accept it — I don’t see how all the empirical observation in the world would get us anywhere.
As far as propositional truths go, it seems to accurately apply to all propositions and all truths. Is A = A itself a propositional truth, the Uber one? I don’t know, but maybe it’s a place to start. No need to get all fancy.
But that definition of logic seems like a circular argument. That’s how we define logical contradictions. But what if we loosened up one our logical laws, as proposed by David Finkelstein? This might eliminate some of the apparent contradictions of quantum physics, and it would show that our rules of logic are also empirical.
https://isidore.co/misc/Physics%20papers%20and%20books/Quantum%20Semiotics/BK%27s%20Texas%20Tech.%20book/References/related%20ref.s%20from%20Dickson%27s%20Non-Rel%20Quant.%20Mech.%20philo.%20article/The%20Logic%20of%20Quantum%20Physics%20(Finkelstein%201962).pdf
Another approach is paraconsistent logics, where a single contradiction does not bring down the whole system. They have practical uses in real-world database systems, and do at least attempt to model some of the the non-catastrophic nature of contradictions in human reasoning.
That may be a reasonable approach in some practical applications, but it doesn’t work in math or in physics theories, because one contradiction can generate an infinite number of contradictions. We believe valid natural laws cannot be self-contradictory, so a contradiction is evidence that such a law is wrong.
Still, physicists have tolerated some kinds of contradictions, for example some of the infinities that arose in quantum electrodynamics.
An interesting aspect of para-consistency which is somewhat like science is that contradictions do have local effects (sometimes dramatic), but generally do not propagate throughout the whole system.
Do people claim that the arts can produce propositional truths? If they do, I would like to see an example. I’m not saying that it does not. I just want to know what they mean.
The arts are at least based on propositional truths. For example, harmonic structures derive from the overtone series,
which is a natural phenomenon. And that natural phenomenon can be described in mathematical terms, as multiples of the base frequency. Does this imply a propositional aspect of mathematics—namely as a systematic way to describe natural, i.e. observed, phenomena?
All witches wear clothes;
Debbie wears clothes;
Therefore Debbie is a witch.
Fallacy of division I believe, correct me if I’m wrong. Might be one of the most common logical fallacies out there.
This is why I still value philosophy…I think it does help to scrub ideas for their logical hygiene. Pure logical consistency may not be a sufficient condition for propositional truth, but it seems to be a necessary one. And a lot of truth propositions out there simply fall at this first hurdle.
And what also floats on water?
/MontyPython
Yes, yes, but of course I said why I valued philosophy in my post. Did you think I was denigrating the value of philosophy in general?
No, apologies if my post implied that. Was not my intent. And one could argue that science also checks for logical consistency so perhaps the usefulness of philosophy has already been subsumed into science.
It seems philosophy and, especially, mathematics are vital tools in the creation of hypothesizes and theories, but until confirmed by empirical experiments, they are abstract ideas.
Mathematics certainly gives propositional truths, for example, “there are infinitely many prime numbers.” One may argue that this is merely a consequence of the axioms of number theory, but I will argue that it is objectively true.
Imagine an activity of constructing rectangular grids from pebbles (including trivial grids in which all pebbles lie on a single line). We begin with two pebbles, then three, four, and so on. Assume we never run out of pebbles as we go from
n pebbles to n+1. We always try to construct a nontrivial grid.
With two and three pebbles, we can construct only trivial grids. With four, there is a nontrivial one: two rows of two pebbles. With five, only a trivial grid is possible. With six, there is again a nontrivial grid: two rows of three pebbles (or three rows of two).
The proposition above says that no matter where we are in this activity, if we continue far enough, we will always reach a stage at which no nontrivial grid can be constructed.
Can any rational being, human or not, really doubt that this proposition is objectively true? Note that my claim does not require the belief that numbers exist in some Platonic realm.
Never don’t beware nested negations.
Thank you for correcting my grammar: “only trivial” instead of “no nontrivial,” right? I quite agree.
Most of this is above my paygrade to contribute to but I’m always impressed by the level of thought and education here.
D.A.
NYC
Re strong (Platonic) mathematical realism, many eminent mathematicians have believed in it, such as Kurt Gödel¹. Others have believed in it early on but later rejected it, such as Hilary Putnam.
What makes this belief so attractive? IMO it’s because working with mathematical objects feels real: when you push them, they push back. There are experiences of something solidly there. They are what they are regardless of how you feel about it. They exist.
There are of course various alternative ontologies. FWIW, I favour “mathematical fictionalism”.
https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Fictionalism
. . . . .
¹ Gödel had the panache to prove the Completeness Theorem for first-order logic, and later on the Incompleteness Theorem. Definitely an out-of-the-box thinker. Sadly, he ended his days as a delusional paranoiac, convinced he was being poisoned and so starved himself to death. (Habitually thinking outside the box risks forgetting where the box is, or even that there ever was a box.)
It is interesting that there are still some proponents of fictionalism around. I suspect they are a dying species, like constructivists in mathematics. Hartry Field’s view that the sense in which, say, the proposition that there are infinitely many prime numbers is “pretty much the same as the sense in which ‘Oliver Twist lived in London’ is true” makes little sense to a mathematician. The former, as I have tried to argue, is objectively true, while the latter is true only “within a story.” There is a huge difference between the two.
There is indeed a difference; ISTM you’re on the wrong side of it. The main problem with strong mathematical realism is the same as with any other sort of Idealism: how does that super-natural world (of Platonic forms / numbers / minds / …) manage to interact with the familiar natural world (of objects / calculations / brains / …).
Descartes’ proposed solution to the mind-body problem was that the brain’s pineal gland is somehow the interface. Well, we don’t know for sure that it isn’t; basically an argument from ignorance, as worthy of respect as the god(s)-of-the-gaps one. And that was the best that René, an extremely clever and knowledgeable guy, could come up with.
You and I agree that there is no largest prime number. The disagreement is whether that proposition is part of the fabric of some other world, or of some constructed narrative in this one. The latter view has the great advantage that we know that such things really exist; the actual existence of the Harry Potter series is not in doubt, the non-existence of Harry Potter himself is too (except maybe for some rabid fans).
1) Bertrand Russell noted that philosophical arguments, when verified, leave philosophy and join science. Philosophy is not quite the graveyard of unanswered questions.
2) It seems to me that 16 divided by 2 equals 8 is a definition based upon other definitions. Indeed, mathematics could be called the working out of definitions. Since context controls definitions, a definition proper in one context is improper in another. Thus the Pythagorean theorem doesn’t hold in non-Euclidean geometry.
What is important is that these definitions are based upon observations of the universe. 1 + 1 equals 2, and 2 + 2 equals 4, set the pace. As such, they can be called propositional truths rather than mathematical truths.
Is working out pi to the millionth place overkill? Could any drawn circle be so accurate? Not if molecules move. Definitions seem to have a breakdown point that requires more observation of the universe for reverification.
How does the distinction between “propositional truths” –– declarative statements about the world that are deemed “true” because they give an accurate description of something in the world or universe–– and other kinds of statements compare to the distinctions between analytic and synthetic or a priori and a posteriori or necessary and contingent?
I took it that Prof. Coyne intended for his distinction to be the same as those you list, which are standard in philosophy. He just used language that was more descriptive, rather than the standard terminology.
Platonism fails with quantum theory. Schrodinger’s and Heisenberg’s equations are both successful descriptions of quantum mechanics, but they are different mathematically. Thus mathematics doesn’t describe an underlying Platonic reality.
Mathematics and philosophy, along with computers and AI, sometimes facilitate the discovery of new propositional truths, although on their own they do not suffice to empirically confirm propositional truths, unlike sensing tool technology which can more directly participate in empirical observation. We who hold this point of view can be labeled philosophical empiricists.
Few people are sticking up for philosophy as opposed to mathematics for producing propositional truths. The prime number contention does give me pause, and it seems a good argument for propositional truths, especially because you can prove it. But can it be falsified? That is, is there a way of showing that the number of primes is NOT infinite?
I typed that falsification question into Google – which I assume those interested will try out – this seems the précis :
” […] any attempt to claim there’s a largest prime or a finite set would lead to a logical contradiction.
[…]
No Valid Counter-Evidence: Any proposed list of “all primes” can be used to generate another, larger prime, showing the list is always incomplete. ”
I also reviewed some Stack Exchange discussion. I’d note from my reviewing that Euclid’s proof is constructive – because there was some confusion on that point.
I’ll stick up for philosophy as producing propositional truths. Specifically, the kind of philosophy advocated by Wilfrid Sellars: explaining the consequences of scientific discoveries in the language of everyday experience (or as Sellars put it, comparing the scientific image with the manifest image).
Some philosophy done in this spirit: essays where Jerry tries to argue that science rules out free will. Of course, you have to draw all your scientific starting points from actual science, and all your manifest image points from everyday experience, to pull it off. It’s harder than it looks. But if done carefully enough, this process would lead to true philosophical conclusions.
I don’t know how to answer the question.
Any mathematical structure whose essential aspects can be exactly modelled by a physical construction will yield the same truths, as long as you don’t cheat. It’s the same thing. The propositional statement about the primes is an example: pebbles and grids model aspects of arithmetic. Statements about numbers can be illustrated using this simple model.
To ask for a way of falsifying the statement is the same as asking if there is a way of showing that the theorem is wrong.
This is distinct from the possible falsification of a statement of physics like Newton’s law of gravity. The mathematical formulation is a proposed model of a natural phenomenon. We did not construct the planetary system as an implementation of Newton’s law.
The discussion above about the Axiom of Choice is similar. Even though we don’t have a physical construction, we may safely say that no human being is going to prove from the Zermelo-Fraenkel that the axiom of choice or its negation is false. That’s because we believe that Godel and Cohen proved that this is not possible. I don’t think the statement is different from saying that no human is going to disprove any proven theorem.
But the discussion is interesting: physics advances by creating mathematical models that get progressively closer to reality. Does this suggest the existence of a model that is an exact representation of nature? If it does, it would seem as if nature is a model of that mathematical structure as much as that mathematical structure being a model of nature.
The problem is, would we ever be able to tell if this were the case? 🙂
I meant this as a response to #18 above.
To ask for a way of falsifying the statement is the same as asking if there is a way of showing that the theorem is wrong. That is, it was never a theorem in the first place. Otherwise it would be a contradiction and would break mathematics 🙂
An errant planet falsifying Newton’s law would not imply an incorrect mathematical statement. It would merely imply that the theory is bad.
“Does this suggest the existence of a model that is an exact representation of nature? If it does, it would seem as if nature is a model of that mathematical structure as much as that mathematical structure being a model of nature.”
Dang, you beat me to it! But yeah, this is why I think some mathematical statements are true.
The argument is valid, meaning it is logically consistent, but it is not sound, meaning one or more of the premises are not true. That is, assuming witches are what we normally take them to be, i.e., supernatural creatures of folklore, then the first premise is not true, since witches don’t exist. In that case, I don’t believe it’s an example of the fallacy of division. On the other hand, if you consider witches to be females that, historically and even today. are deemed to be “witches” either by others or themselves, then it could be a fallacy of division.
Note: this comment was intended for the witches argument.
The argument is indeed invalid, as you can see immediately by restating the first premise as “All airline pilots wear clothes.”
Then the conclusion, “Debbie must be an airline pilot” (because she wears clothes) is clearly absurd even though all the nouns in the syllogism undoubtedly exist. The argument would be valid only if the minor premise was, “Debbie is an airline pilot,” and the conclusion was, “Debbie must wear clothes.”
A valid, though unsound, argument about witches would be:
“All women who float in water are witches.
Debbie, a woman, floats in water.
Therefore Debbie must be a witch.”
Note that “All witches float in water” as the major premise would make the conclusion invalid again. It would have to be “Only witches float in water,” to restore validity. Also valid would be to say as the major premise, “All witches float in water”, and as the minor premise, “Debbie sank and drowned.” Then the conclusion, “Debbie, God rest her immortal soul, was not a witch,” would be valid.
Thank you for your well-explained response. I stand corrected.
Most gracious of you. You’re welcome.
I’m looking forward to your article
Jerry, you define “propositional truths” this way:
‘“propositional truths”, that is, declarative statements about the world that are deemed “true” because they give an accurate description of something in the world or universe.’
Here’s a logical truth: “For any X, X is X.”
This logical truth meets your own definition of “propositional truth,” namely, it is a declarative statement that gives an accurate description of something (indeed, all things, even all possible things) in the world or universe.
So, the logical truth stated above is a propositional truth.
Logic is a part of philosophy.
So, philosophy has propositional truths.
You then say later:
“All As are B; x is an A; therefore x is B—doesn’t depend on the content of A and B: it’s a logical truth.”
“Again, the statement is indeed a logical truth, but not a propositional truth because it cannot be tested to see if it’s true or false. Nor, without specifying exactly what A and B is, can the empirical truth of this statement be judged. I claim that all philosophical “truths”—logical truths without empirical input—are of this type.”
Suddenly, “propositional truth” now means being capable of being “tested to see if it is true or false.”
But this definition is not equivalent to your first given definition of “propositional truth.”
So which is it?
Philosophy teaches us to be consistent in our definitions of terms.
Second, both logical propositions, “For any X, X is X” and “If ‘All As are B’; and if ‘x is an A’; therefore, ‘x is B’”, can be tested to see if they are true or false. Just plug in terms for the variables, and you will see that these logical propositions are true. So, logical truths are propositional truths even on your second definition of “propositional truth.”
To conclude, neither of your definitions of “propositional truth” successfully excludes logical truths. (I could give parallel arguments for mathematical truths.)