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May 8, 2008

Pernicious Symbolization

Posted by David Corfield

Gian-Carlo Rota upset a number of analytic philosophers when in The pernicious influence of mathematics upon philosophy he likened their use of symbolism to someone paying for groceries with Monopoly money. But it didn’t take an outsider to object to such practices. Gilbert Ryle, one of the so-called ‘ordinary language philosophers’, reviewing Rudolf Carnap’s Meaning and Necessity in Philosophy XXIV, 1949, remarks on Carnap’s

…growing willingness to present his views in quite generous rations of English prose. He still likes to construct artificial ‘languages’ (which are not languages but codes), and he still interlards his formulae with unhandy because, for English speakers, unsayable Gothic letters. But the expository importance of these encoded formulae seems to be dwindling. Indeed I cannot satisfy myself that they have more than a ritual value. They do not function as a sieve against vagueness, ambiguity or sheer confusion, and they are not used for the abbreviation or formalization of proofs. Calculi without calculations seem to be gratuitous algebra. Nor, where explicitness is the desideratum, is shorthand a good substitute.

(Interlard is literally ‘to intersperse with alternate layers of lard’.)

We owe at least to Carnap our term functor, if not its meaning. But here’s Ryle again:

He likes to coin words ending in ‘…tor’. He speaks of ‘descriptors’ instead of descriptions’, ‘predicators’ instead of ‘predicates’, ‘functors’ instead of ‘functions’, and toys with the project of piling on the agony with ‘conceptor’, ‘abstractor’, ‘individuator’, and so on. But as his two cardinal words ‘designator’ and ‘predicator’ are employed with, if possible, even greater ambiguity and vagueness than has traditionally attached to the words ‘term’ and ‘predicate’, I hope that future exercises in logical nomenclature will be concentrated less on the terminations than on the offices of our titles.

Well, at least we’re assured of the worth of the offices of ‘associator’, ‘Jacobiator’ and ‘Jaciobiatorator’, even if the nomenclature is questionable.

Posted at May 8, 2008 9:46 AM UTC

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Re: Pernicious Symbolization

I’m just glad no category theorist has felt the need for “modulator”, as in Marvin the Martian with his pernicious “Illudium Q-36 explosive space modulator”!

On a slightly more serious note – I will never forget seeing Rota give a talk on the theme of his article The Pernicious Influence of Mathematics on Philosophy. When the mike was opened for questions and comments, an audience member (evidently a philosophy professor) came up and confessed that he didn’t know what Rota was talking about, and could Rota furnish just one example of a philosopher committing these alleged sins? Rota flashed a grin and said merely, “I wish it were as you say,” or something equally lame. The professor apparently couldn’t believe his ears (like, “Is that it?!”), and was asked if he’d like to say anything to that. At first he was so taken aback he couldn’t think of a thing, and then finally shot back, “Your wish is granted!”

Somehow that little exchange has become emblematic for me of my feelings about Rota’s Indiscrete Thoughts (where the article is reproduced): Rota makes some very interesting points, but then proceeds to ruin them through exaggeration or rhetorical overkill or things said in very bad taste. I also have the impression that a lot of his factual assertions there will not stand up to scrutiny, providing his enemies with some easy marks (although I wish I had made notes in the margins to back that up).

Posted by: Todd Trimble on May 8, 2008 5:48 PM | Permalink | Reply to this

Re: Pernicious Symbolization

There is a class of philosophical problems which are susceptible to some form of mathematization. However there is even a larger class of problems self-inflicted from our misunderstanding the use of language. The real fun begins when you mistake one class for another and create a whole new class of problems!

Posted by: Paul on May 8, 2008 6:36 PM | Permalink | Reply to this

Re: Pernicious Symbolization

Here’s a question then. What characterizes the class of philosophical problems susceptible to mathematization?

Posted by: David Corfield on May 10, 2008 7:48 PM | Permalink | Reply to this

Biology and Culture; Re: Pernicious Symbolization

I strongly suspect that this depends on the culture in which the Philosopher is embedded.

Mayans had good enough Math to solve problems in Astronomy connecting the periods of the Moon and Venus, but little of the depth of this trickled into their cyclic catastrophism.

Archimedes, as we recently learned, could handle Infinity as a number, through Combinatorics. But he had no good way to communicate this to Greek philosophers.

Roman Numerals would not have helped Aristotle if he wanted to mathematize his theory of Poetics, or of Rhetoric.

Lull made a real and early contribution, as did Liebniz.

But it took Boole and Cantor and Post and Turing and Kleene and Shannon and Godel to break into the big time.

More deeply, doesn’t the biology of the organism, and the physical environment in which it is embedded, affect this issue?

Dolphins can’t count on their fingers, so have no “rules of thumb.” But they do acoustic holography, it seems.

Science Fiction is filled with examples of intelligent creatures encoding deep ideas in math peculiar to their physiology and neuroanatomy.

Greg Egan has given considerable thought to this, and produced works of stunning metaphysical originality, which are fun to read – and almost impossible not to think about later.

Posted by: Jonathan Vos Post on May 10, 2008 9:14 PM | Permalink | Reply to this

Re: Pernicious Symbolization

The problems that are solvable get solved. I don’t think there is a preemptive way to determine other than one’s own intuition.

When George Boole was working on logic he considered it philosophy and not mathematics. Yet nowadays Boolean logic and algebra are purely mathematical topics. Perhaps it is better to rephrase the question to “What philosophical problems are really mathematical ones?”.

The answer being.. the ones that are solved mathematically!

Posted by: Paul on May 12, 2008 5:06 PM | Permalink | Reply to this

Re: Pernicious Symbolization

This seems like an unanswerable question. The philosophical problems that are susceptible to mathematisation are just those that some smart researcher somewhere has figured out how to mathematise. Before fields like formal logic or mathematical physics were opened up nobody fully realised that reasoning, or the motion of bodies, could be made mathematical, and I doubt that any classification we could give now could predict what some future researcher might invent.

Posted by: Dan P on May 12, 2008 5:23 PM | Permalink | Reply to this

Re: Pernicious Symbolization

I wasn’t after predictions. What intrigues me is what happens to disciplinary boundaries as topics become mathematized.

Take something like ‘time’, a topic which became thoroughly mathematicized by physicists, and yet is still treated by philosophers looking for interpretations of relativity theory and quantum gravity. But ‘time’ is also treated by philosophers without a scientific bone in their body in terms of our lived temporal experience. One might argue that the concept of ‘time’ has merely split.

Something which may experience a similar fate is ‘causality’. If Judea Pearl wins out, it will become another scientific topic treated via the apparatus of graphical models and the ‘do’ calculus. But we may also retain a more phenomenological study.

With some topics philosophers don’t just play the role of interpreters, but can get stuck into the development of a calculus, e.g., blending probability theory and logic. But maybe if they succeed it will soon no longer be within their remit to develop this calculus further.

What seems a bit odd to me is that, in the Anglophone world at least, there’s such little encouragement for interpretative work on mathematical topics: symmetry, space, quantity, equivalence. Could working on Klein 2-geometry be construed as philosophical?

Or if you feel it needs an applied discipline for useful interpretative work, why is so little done on topics from computer science - state, process, algorithm?

I rather doubt that what happens to topics as they are mathematized, with regard to their status as being worthy of philosophical attention, does so in a principled way.

Posted by: David Corfield on May 12, 2008 6:20 PM | Permalink | Reply to this

Re: Pernicious Symbolization

> Or if you feel it needs an applied discipline for useful interpretative work, why is so little done on topics from computer science

I’m trying to get a handle on what exactly ‘interpretative’ might mean in these contexts. It seems to me that writings in these areas come with their interpretation. What more interpretation could there be? What kind of interpretation are you thinking of?

In fact, maybe you could point to some non-Anglophone work that illustrates what you mean. (In English translation, or maybe I could muddle by in French.)

> One might argue that the concept of ‘time’ has merely split.

Funny you pick the example of time. I’ve been trying (with the emphasis on trying) to read some Heidegger in recent weeks. The path from “non-scientific” time to “scientific” time is one of his main topics, if not the main topic. But I’m not confident to actually say anything about what he says.

Posted by: Dan P on May 12, 2008 9:07 PM | Permalink | Reply to this

Re: Pernicious Symbolization

I’m trying to get a handle on what exactly ‘interpretative’ might mean in these contexts. It seems to me that writings in these areas come with their interpretation. What more interpretation could there be? What kind of interpretation are you thinking of?

I’m not saying the interpretative activity of practitioner and philosopher are disjoint. In fact at times they are indistinguishable. But you would imagine that each would bring different resources to the discussion.

My question was why this co-operation should happen quite readily in physics, for instance Physics Meets Philosophy at the Planck Scale: Contemporary Theories in Quantum Gravity, but much less so in other disciplines.

My ‘Anglophone’ comment was to point out that things are a little freer elsewhere. E.g., it’s hard to think of a US/UK philosopher resembling Jean Petitot, or one contributing to a book such as What is Geometry?.

The funny thing about your reading Heidegger is that the kind of philosopher who works on physical theories of time is very unlikely to think of reaching out to such resources. Practitioners, it seems to me, are open to a wider range of thinking.

Posted by: David Corfield on May 13, 2008 9:13 AM | Permalink | Reply to this

Math departments versus Philosophy Departments; Re: Pernicious Symbolization

I have, at several secondary and post-secondary schools, been caught in the middle of a turf battle between Math departments and Philosophy Departments.

The ostensible prize is: who gets to teach Introduction to Logic? It seems to usually be decided by historical precedent: whichever department first taught it.

I had taken courses in Propositional and and Predicate Calculus at Stuyvesant High School, using university textbooks.

At Caltech, I continued with Infinitary Logic and with Model Theory, despite lacking the formal prerequisites. I simply insisted “but I am POST!” and the professor let me in, and not until mid-semester asked exactly how I was related to Emil Post. I said that I was not; and that he had been fooled by illogic.

When I was an Adjunct Prof of Math at Woodbury University, I expanded the Logic component of Intermediate Algebra as taught, using Lewis Carroll puzzles, which the students loved.

But the administration would not let me expand this further, as the Philosophy Department traditionally taught Intro Logic, and taught it quite badly.

I repurposed the Lewis Carroll worksheets I’d typed when I taught remedial algebra in a Pasadena High School. But they would not entertain a full course on the subject. After all, what do mathematicians with degrees also in Computer Science, and publications in Physics and Biology, know about Logic?

The battle raged one way in Great Britain, with my teachers’teachers’ teachers’ and their teachers in several branches.


John McTaggart Ellis McTaggart [1866-1925] was a Fellow of Trinity College, Lecturer in Moral Sciences, and a Nonreductionist. He was the author of “Studies in Hegelian Cosmology. The Philosophy of Hegel”
[Dissertation, 1898; 1901; Garland, 1984].

This work explored application of a priori conclusions derived from the investigation of pure thought to
empirically-known subject matter; human immortality; the absolute; the supreme good and the moral criticism; punishment; sin; and the conception of society as an organism. McTaggart was controversial for claiming that time was unreal: “The Nature of Existence” [Cambridge University Press, 1921]; “The Unreality of Time” [Mind, vol. XVII].

G. E. Moore (George Edward Moore) [4 Nov 1873-24 Oct 1958] was an English Realist Philosopher, and a leading anti-Idealist. His major publications include “Principia Ethica” [Cambridge, 1903; 1993];
“Ethics” [London, 1912]; Philosophical Studies [London, 1922];
“Some Main Problems of Philosophy” [London, 1953]; Philosophical Papers”
[London, 1959]; and various posthumously edited volumes.

Bertrand Russell [1872-1970] was the great Philosopher/Logician who studied with Wittgenstein and G. E. Moore, wrote “Principia Mathematica” with Alfred North Whitehead, had a major impact on Analytic Philosophy, was a best-selling popular author, was jailed for anti-war protests,
and won the Nobel Prize (1950, Literature).

Norbert Wiener [1897-March 1964] was the creator of Cybernetics, who studied under G. H. Hardy and Bertrand Russell by fellowship. He received his Ph.D. in the Philosophy of Mathematics at Harvard at age 18, under Josiah Royce [1855-1916] – the foremost American Idealist; under
Philosopher/Poet George Santayana [1863-1952]. Norbert Wiener then studied at Gottingen under David Hilbert [23 Jan 1862-14 Feb 1943].

I am a student of a student of Norbert Wiener, the intermediate link being a huckster palgiarist whom I prefer not to discuss in open forum.

In some other threads of n-Category cafe I have alluded to the importance of the work of

See today’s preprint:

Paraconsistent First-Order Logic LP# with infinite hierarchy levels of contradiction
Authors: Jaykov Foukzon
Subjects: Logic (math.LO)

In this paper paraconsistent first-order logic LP# with infinite hierarchy levels of contradiction is proposed. Corresponding paraconsistent set theory KSth# is discussed.

The text of the paper begins:

“The real history of non-Aristotelian logic begins on May 18,1910 when N.A. Vasiliev presented to the Kazan University faculty a lecture ”On Partial Judgements,
the Triangle of Opposition, the Law of Excluded Fourth” [Vasiliev 1910] to satisfy the requirements for obtaining the title of privat-dozent. In this lecture
Vasiliev expounded for the first time the key principles of non-Aristotelian, imaginary, logic.”

Posted by: Jonathan Vos Post on May 13, 2008 7:57 PM | Permalink | Reply to this

Re: Math departments versus Philosophy Departments; Re: Pernicious Symbolization
Paraconsistent second order arithmetic Z#2 based on the paraconsistent logic LP# with infinite hierarchy levels of contradiction. Berry’s and Richard’s inconsistent numbers within Z#2
Authors: Jaykov Foukzon
(Submitted on 18 Jun 2009)

Abstract: In this paper paraconsistent second order arithmetic Z#2 with unrestricted comprehension scheme is proposed. We outline the development of certain portions of paraconsistent mathematics within paraconsistent second order arithmetic Z#2.In particular we defined infinite hierarchy Berry’s and Richard’s inconsistent numbers as elements of the paraconsistent field R#.

Posted by: Jaykov Foukzon on June 19, 2009 11:56 AM | Permalink | Reply to this

Re: Pernicious Symbolization

David Corfield wrote:
“Something which may experience a similar
fate is ‘causality’. If Judea Pearl wins
out, it will become another scientific
topic treated via the apparatus of
graphical models and the ‘do’ calculus.
But we may also retain a more
phenomenological study.”

Judea Pearl wrote:
“Considering the organization of scientific
knowledge, it makes prefect sense that we
permit scientists to articulate what they
know in plain causal expressions, and not
force them to compromise reliability by
converting to the “higher level” language
of prior probabilities, conditional
independence and other cognitively
unfriendly terminology.”

Stephen wrote:
Pearl’s description seems compatible with the goal of pragmatically minimizing “Pernicious Symbolization”. In your book I didn’t notice a criticism of Pearl, although Jon Williamson, your co-editor, pushes epistemic causality elsewhere. I’m not sure what the scope of your “phenomenological study” encompasses. Just what you mean by that? I did find what appears to be a related philosophical question:

“A crucial question for those interested
in the semantic and metaphysical
foundations of the structural equations
framework is the status of the
counterfactuals encoded by the structural
equations. Are they semantically and
metaphysically primitive so that the
structural equations are simply a summary
of the more basic counterfactuals? Or are
the structural equations themselves to be
taken as the conceptual and metaphysical
primitives, with the counterfactuals
having a secondary, derivative status?”
(Woodward and Hitchcock adaption of Pearl)

Posted by: Stephen Harris on May 15, 2008 12:00 PM | Permalink | Reply to this

Re: Pernicious Symbolization

It may well be that the study of time fractures more readily into separate streams. The scientific study concerns the story from absolute time to relativistic notions to who knows what if we ever get to a quantum gravity. Meanwhile there will be those who wish to describe our experience of time in a ‘phenomenology’ of time. It’s plausible to think that developments in the former won’t affect the latter.

One big debate about cause at the moment is whether to treat it monolithically, or pluralistically. The latter position takes the notion ‘cause’ to be at work in a range of different disciplines, and warns us from thinking that their should be a common treatment, or that one should be reducible to a primary physical notion. The other camp hope to treat cause uniformally, perhaps through counterfactual analysis (what would happen were something else to happen) or through a continual transfer of some quantity, say, energy.

I tend to side with the pluralists. I doubt any case where we would want to use ‘cause’ can be analysed properly as a causal network.

Posted by: David Corfield on May 16, 2008 1:26 PM | Permalink | Reply to this

Time is continuous, discerete? Re: Pernicious Symbolization

Some glitch in the n-Catgory Cafe web site has stopped me twice from
posting this on 16 May 2008.

Pernicious Symbolism

The first question, both philosophically and experimentally – is time continuous (classical Physics) or discrete/quantized?

One approach leads to Zeno’s paradoxes; the other is favored in, for instance, Loop Quantum Gravity.

Demonstrate How Information Can Escape From Black Holes

“space-time is not a continuum as physicists once believed. Instead, it is made up of individual building blocks, just as a piece of fabric, though it appears to be continuous, is made up of individual threads….”

It would help acceptance of the theory of the controversial Peter Lynds if a flaw can be found in the work announced above.

Peter Lynds would naturally think that the most obvious flaw is that space-time isn’t discontinuous. He just checked, and verifies that Ashtekar does work on LQG.

The next question is the phenomenology of human perception of time.

See, for example, equally recent, the periodic table of visual illusions and a striking conclusion by one research group:

(Eye) Ball: Visual System Equipped With ‘Future Seeing Powers’

Posted by: Jonathan Vos Post on May 18, 2008 5:50 PM | Permalink | Reply to this

Inconsistent Countable Set in Second Order ZFC

British Journal of Mathematics & Computer Science, ISSN: 2231-0851,Vol.: 9, Issue.: 5

In this article we derived an important example of the inconsistent countable set in second order ZFC (ZFC2) with the full second-order semantics. Main results: (i) ¬Con(ZFC2), (ii) let k be an inaccessible cardinal and Hk is a set of all sets having hereditary size less then k, then ¬Con(ZFC + (V = H_k)).

Posted by: Jaykov Foukzon on February 15, 2017 11:56 PM | Permalink | Reply to this

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