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 inmhy 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.
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.
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.]
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!
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
😁
“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.
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.
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
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.