On extending mathematical attitudes to natural languages

Kris Brown - Inaugural Tutorial Day 2023


(press s for speaker notes)

7/25/23

Introduction

$$

$$

Artificial languages (including mathematical languages) have a different flavor from natural languages.

We tend to like some artificial languages (like mathematics).

• Deductive reasoning
• Compositionality (\(\phi\)’s meaning just depends on its syntactic structure and the meanings of its simple parts.)

• Upfront, explicit definitions + premises
• Monotonicity (once we are justified in saying something, we will forever continue to be justified in saying it)

Like Plato, we see them as an ideal to which other languages are to be held. Other discourse is inadequate insofar as it does not have these properties. Why? It would be nice if …

• reasoning could be a mechanical, thoughtless process.
• context had no impact on our utterances’ validity.

• meaning were transparent / we had absolute certainty about some things.
• one need not worry about justification of established facts / we had a feeling of progress towards something Eternal.

[Title slide] Hi so I want to talk about something that’s only math-adjacent. Mostly stuff I think is fun but also has potential to be relevent to Topos.

The central theme is that natural languages are quite different from artificial languages. (seems obvious but I’ll argue it’s a deep point)

(can always ask a definition or, if no further definitions, what its semantics is)

. . .

inadequate candidates: morality, astrology, art critics, poetry (dare I say, epistemologically bankrupt)

(bottom four bullet points correspond to those above)

This is my psychologization of why people want to hold other discourse to this standard. (This is something people, myself included, are easily tempted to do.) But I’ll argue that this is not a good thing to do uncritically and not in the spirit of category theory.

Overview

Formal language	Natural language
Logical inference	Material inference
Monotonic logic	Nonmonotonic logic
Logical empiricism	Pragmatism
Atomism about meaning	Holism about meaning
Language/theory distinction	The ‘vocabulary’ vocabulary
Truth	Justification
Semantics	Pragmatics
Representationalism	Inferentialism

So this is my summary of a lot of differences between formal and natural languages. Something might be appropriate for formal languages that isn’t appropriate for natural languages, and so we need to be careful to not use the wrong tool for the job.

I’ll just jump into some of these differences and maybe at the end we can take a fresh look at this table.

Material inference

• If I strike this match, it will light.
• If it’s red, then it’s colored

• If the streets are wet, it recently rained.
• If NH is to the East of CA, then CA is to the West of NH.

	Material inference	Logically-valid inference
Def	Can be changed from a good inference into a bad one by altering some nonlogical vocabulary.	True no matter what you plug in for the variables or substitute for the non-logical vocabulary.
Slogan	Descriptive terms appear essentially	Descriptive terms appear vacuously
Form	True, but not because of its logical form	True in virtue of its logical form

1. There are some inferences that are good, not in virtue of their logical form.
2. Rather than thinking the descriptive terms (“match”, “red”) have some essence which has the suppressed premises, we can understand the content of these descriptive terms in terms of the materially good inferences they appear in (as premises or conclusions).

Material proprieties of inference are more fundamental / conceptually prior.

Need the notion of a good inference in order to understand what a logically good inference is.

These are examples of material inferences, an interesting concept introduced by Wilfrid Sellars. They may look funny to you because they don’t seem valid or universally true. That may be because you’re thinking of logical inferences (i.e. tautologies).

. . .

(Read the table) There is a kind of battle of the conceptual priority of the two. If you are a fan of formal logic, you might try to understand the first column in terms of the second. (you want the reason they’re true to be something you can mechanically compute, because of what you think is an ideal). When you ultimately try to do that order of explanation, you’ll start becoming judgmental of these good inferences, seeing them as inadequate, surely they’re suppressing hidden premises (such as there being oxygen in the room for the match one). This is precisely the kind of attitude I talked about in the first slide. I’ll argue the order of understanding should actually be the other way around for natural languages.

. . .

Once you have the concept of ‘material inference’ in your head, you’re led to two good ideas:

It also generalizes beyond “logically valid” but you can take for any subset of your vocabulary and saying sentences which are true in virtue of their, e.g. theological form (where we aren’t allowed to substitute the word “pious”).

Nonmonotonic logic

1. “If I rub this match along the striking surface, then it will ignite.” (\(p\rightarrow q\))
2. “If \(p\), but the match is inside a strong electromagnetic field, then it will not ignite.” (\(p\land r\rightarrow \neg q\))
3. “If \(p\) and \(r\), but the match is in a Faraday cage, then it will ignite.” (\(p\land r\land s\rightarrow q\))
4. “If \(p\) and \(r\) and \(s\), but there is no oxygen in the room, then the match will not ignite.” (\(p\land r\land s\land t\rightarrow \neg q\))
5. …

• Categorical semantics is open problem (MathOverflow)

How these new ideas might be in the ‘spirit’ of category theory:

• Understanding the conceptual content of “bat” as defined by its interconnections in networks of material inference (rather than those being the proper inferences to make in virtue of the nature/Platonic idea of “bat”)
  • This is evocative of ‘morphisms-first’ approach to conceptual content.
• Modeling the openness to change endogenously rather than as a meta-level attitude
  • This is very in spirit with open systems vs closed systems.

So material inferences, unlike classical logical inferences, have a non-monotonic logic structure.

Reminds me of regnets.

Logical empiricism

How might someone with an affinity for formal logic approach the problem of natural languages? Consider a tradition in philosophy called logical empiricism:

• We have freedom to pick our language (e.g. “bachelor := unmarried man”)

principle of tolerance: we are not in the business of setting up prohibitions but of arriving at conventions… In logic there are no morals. Everyone is welcome to set up his logic, i.e., his form of language, as he pleases. If he wants to discuss it with us, though, he needs to give syntactical specifications rather than philosophical debates.

— Carnap, Logical Syntax of Language §17

• Once we do this, the world settles what our terms refer to and which theory (i.e. set of true sentences) is correct.

This language/theory distinction is characteristic of artificial languages. Confusion stems from applying this model to natural languages, where the analogies (e.g. deduction, premise) don’t exactly work the same way.

So what is in opposition to the picture I painted?

So Carnap here is being a kind of pragmatist. Now I know I contrasted pragmatism with logical empiricism in the previous slide, so I should clarify that he’s being a pragmatist about our choice of language. So the epidemiologist, the mathematician, the astrologist, the utilitarian, they all get to pick (whether they know it or not) some grammar and terms with their definitions etc. There’s nothing other than what we eventually find to be useful in guiding us here.

. . .

For example we might have some sort of scientific language where we declare for all masses, forces, and accelerations, \(F=ma\). We then look to the world to see whether that is true.

So what are the criticisms of this?

In the Aufbau, Carnap heroically attempts to translate statements of the form “Quality \(q\) is at point-instant \(x;y;z;t\).” into a primitive sense data language (+ set theory). Unsuccessful.

Agrippan / Münchhausen trilemma

Assume a transitive structure of justification on set of beliefs.

Then, any knowledge requires a choice:

1. Infinitely regressive argument: the chain of justification goes on ad infinitum.
2. Circular / coherentist argument: justification is not acyclic
3. Foundationalist / dogmatic argument: there exist self-justifying beliefs

\[...\implies p_{i-1} \implies p_i \implies p_{i+1} \implies ... \]

Challenge for logical empiricism

	Belief	Inference
Regress stopper	Sense data	Analytic / logical inference

But for something to be a regress stopper, it must be justified independent of any other collateral commitments.

(Read the trilemma) Note that another option is to take a non-monotonic logic approach and consider truth to be relative to the arguments that are currently in play, being open to extending to as-of-yet unknown defeasors.

. . . So logical empiricism is going to take option 3.

Surprisingly there are two kinds of regresses when we try to bring the artificial language deductive style of reasoning to the natural language setting. The obvious one is the regress of premises, what belief justifies my belief \(p_i\), well that’s some other belief, \(p_{i-1}\), and what justifies that? The other regress concerns my justification in going from \(p_i\) to \(p_{i+1}\) - what obliges or entitles me to do that? Some reason is given, and now the question is why does that reason oblige me to accept the previous obligation, and so on.

The idea of sense data as the foundation for our beliefs and analytic / definitional truth as the foundation for our inference is very powerful! It took some two brilliant minds to help us see that these things cannot actually play the role of being unjustified justifiers since they cannot be understood as operating independent of our entire network of understanding as a whole. So the arguments against this are going to be arguments from a holism about meaning.

Quine’s Two Dogmas of Empiricism

Analytic truths	Synthetic truths
Statements which are ‘true by definition’.	Require more than definitions to be true.
Doctors who work on eyes are doctors. People who run move.	Doctors who work on eyes are rich. People who run damage their bodies.

Quine looks at various characterizations of analyticity and finds them all inadequate (ultimately circular notions: self-contradictory, synonomy, necessity).

Each statement is not made (atomistically) true or false by the world. The meaning of \(\phi\) is dependent on our context \(\Gamma\) of background assumptions - only the whole constellation faces the tribunal of sense experience.

• When we make a scientific measurement to falsify some theory, the interpretation of that measurement is made in some other theory. The world doesn’t falsify a theory, rather we have preferences of which beliefs we want to keep fixed and which we are willing to update.
• We are free to choose our logic!

[Carnap’s] pragmatism leaves off at the imagined boundary between the analytic and the synthetic. In repudiating such a boundary I espouse a more thorough pragmatism.

This very intuitive distinction played a big role in early Modern philosophy, where it seemed like we definitely knew analytic truths but it was a philosophical challenge to see how we could come to know synthetic truths, especially synthetic a priori truths.

. . .

Here is a gist of Quine’s argument in favor for holism in meaning. This is called the Duhem Quine point (Duhem brought it up in the context of science, but Quine radicalized it to all aspects of knowledge, pointing out that we can even change our rules of logic if it helps us better cope with the world, which is a CT friendly notion).

The totality of our so-called knowledge or beliefs, from the most casual matters of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man- made fabric which impinges on experience only along the edges. Or, to change the figure, total science is like a field of force whose boundary conditions are experience. A conflict with experience at the periphery occasions readjustments in the interior of the field. Truth values have to be redistributed over some of our statements.

Sellars’ Myth of the Given

The narrative being attacked is a process like this:

1. Physical objects -> 2. Sense data -> 3. Noninferential beliefs -> 4. Inferential beliefs

Reductivist empiricism seeks a foundation for knowledge and takes sense-impressions as the unjustified justifiers. Against this, Sellars argues that:

• For a sensory experience to play an evidential role (to serve as the justification of some other claim), it has to be conceptually articulated.
• You can’t just have this concept in isolation, because having the concept is understanding the concept’s inferential relations (holism about conceptual content).

In characterizing an episode or a state as knowing, we are not giving an empirical description of it. We are placing it in the logical space of reasons, of justifying and being able to justify what one says. — Empiricism and the Philosophy of Mind §36

So Sellars’ holism about meaning is influenced by what one has to do to count as grasping a concept. He argues that “grasp of a concept is mastery of a word” - that it is not just in virtue of the fact someone has mastered the use of a word that we become aware that a special mental event of grasping has occurred in their brain - but rather that this is what grasping consists in.

Attacking logical empiricism

	Sellars	Quine
Myth	Myth of the Given	Myth of the Museum
Relevant work	Empiricism and the Philosophy of Mind (1956)	Two Dogmas of Empiricism (1951)
Concept challenged	Sense-givenness	Meaning-givenness
Against atomism in…	Epistemology	Semantics
Kantian representation	Sense impression	Meaning-relation
Regress stopped	premises	inferences

The simplest way to describe the common features of Quine’s and Sellars’s attacks on logical empiricism is to say that both raise behaviorist questions about the epistemic privilege which logical empiricism claims for certain assertions, qua reports of privileged representations. — Rorty, Philosophy and the Mirror of Nature, Chapter 4

Quine (resp. Sellars) asks “What do you have to be able to do?” in semantics (resp. epistemology) in order to argue for his holism.

So to summarize this attack on this language-theory distinction by Sellars and Quine.

. . .

Rorty was an intellectual historian writing around 1980, looking back on what happened to logical empiricism and had the eyes to see a striking parallelism and duality to the two critiques.

Alternative to the language-theory distinction

The ‘vocabulary’ vocabulary. Meanings and facts come together at the same time.

Rather than search for a fixed foundational vocabulary to deduce truth from, we ought acknowledge our power of redescription - to change our vocabulary to suit our interests.

• E.g. Convincing someone who has defined a product of sets to redescribe this as the product in Set. It’s not an argument.
• Redescription is rational but not logical: you can rationally persuade (not prove) someone who thinks 1 ought be (defined to be) a prime number.
• No real difference between learning phlogiston (oxygen) has very different properties than we thought it had vs declaring phlogiston doesn’t (objectively) exist.

E.g. the vocabulary of 16th century theology, of modernity, of monsters under your bed.

Point 3: it’s a red herring to worry about do we have the right vocabulary that correctly represents the world as it objectively is. Rather it’s a very valid question whether in the vocabulary of natural science something has the status of existing.

These pointless ontological problems that make people think philosophy is completely out of touch with real problems can be traced to the language-theory distinction, the idea that language in general needs to represent the world (as opposed to the idiom of natural science, which does have the job of representing the world).

Alternative to the language-theory distinction

For certain aims, we want a fixed vocabulary, for reasoning to be monotonic. In mature sciences, a lot of work is taken to allow for discourse to proceed as if the vocabulary were fixed:

• Discursive “clean rooms”, maintained through heroic social disciplinary measures.
• For mature sciences: e.g. in the history of temperature, each time a new way of measuring temperature was discovered, the concept changed.

We may not want to take this attitude towards all forms of discourse, e.g.  moral discourse.

Even scientific discourse we want periodic redescription (Kuhnian paradigm shifts).

E.g. The Enlightment as a redescription of mankind as needing to look outside for moral guidance into looking inside.

Takeaways

• Material (logical) inference, (non) monotonic logic
• Holism vs atomism about conceptual content

• The vocabulary vocabulary vs the language / theory distinction

• David paints a beautiful picture in “What are we tracking?” of good math as crystallizing a phenomenon.

  • ✅ Formalizing a human practice makes it explicit, exposing it to rational criticism.

  • ❌ Thinking of an abstraction as ‘right’ crystalizes it - making it immune to revision.

• ✅ Compositionality / monotonicity are nice properties to aspire for.

  • ❌ But they are inherently atomistic, rather than holistic.

• Determining who Topos works with

• Attitude towards the subject matter experts we work with

Need to be careful in what we think finding the ‘right abstraction’ means

SME comes in and paints a very complex picture of the world. We think of an abstraction that almost works but when there’s a conflict, we are tempted to say that the SME ought do things differently, after all the math is pointing us in a direction, it feels like. One lesson of the talk is just to be cautious or critical of our urge to say that - no doubt some times it is justified!

Achilles and the Tortoise: regress of inferences

The Tortoise accepts a proposition \(p\) and \(p \implies q\).

• The exact \(p\) and \(q\) aren’t important to the moral of the story, though
• The Tortoise is acting like: I’ll do anything you tell me to do, so long as you make explicit the rule you’re asking me to follow.

Achilles tries to convince the Tortoise to accept \(q\) (logic obliges you to acknowledge \(q\)).

The Tortoise is willing to go along with this but demands that this rule be made explicit: so Achilles adds an extra axiom: \(p \land (p \implies q) \implies q\).

Achilles says that, now, you really have to accept \(q\), given that you’re committed to:

• \(p\)

• \(p \implies q\)

• \(p \land (p \implies q) \implies q\).

But the Tortoise notes that, if taking those three propositions and concluding \(q\) is really something logic obliges one to do, then it bears writing down: \[p \land (p \implies q) \land (p \land (p \implies q)) \implies q\]

This can go ad infinitum; the Tortoise wins.

• in any particular case, you can substitute a rule (that tells you you can go from this to that) with an axiom.
• But there have got to be some moves you can make without having to explicitly license them by a principle.
• I.e. you’ve got to distinguish between
  1. premises from which to reason
  2. principles in accordance with which to reason.

Myth of the Given

The narrative being attacked is a process like this:

1. Physical objects
2. Sensing of sense contents
3. Noninferential beliefs
4. Inferential beliefs

• (1->2) Because there is a red object with a triangular facing surface in front of me, I find myself with a sensing of red-triangular content.

The first inference is a causal notion (studied by neurophysiologists).

It is a matter-of-factual relation. It is describable in non-normative vocabulary.

Myth of the Given

The narrative being attacked is a process like this:

1. Physical objects
2. Sensing of sense contents
3. Noninferential beliefs
4. Inferential beliefs

• (1->2) Because there is a red object with a triangular facing surface in front of me, I find myself with a sensing of red-triangular content.
• (2->3) ?
• (3->4) Because I have this belief (along with others), I am justified in believing there is a yield sign in front of me.

The third inference is epistemic, an inferential notion which relates sententially-structured beliefs/believables which are repeatable abstracta. This is a matter of reasons rather than causes. It is not a natural justificatory relation but rather a normative one; the logician rather than the scientist has the final say in adjudicating it.

Myth of the Given

The narrative being attacked is a process like this:

1. Physical objects
2. Sensing of sense contents
3. Noninferential beliefs
4. Inferential beliefs

• (1->2) Because there is a red object with a triangular facing surface in front of me, I find myself with a sensing of red-triangular content.

• (2->3) Because I have such as sense content that I acquire the noninferential belief that there is a red-triangular object in front of me.

• (3->4) Because I have this belief (along with others), I am justified in believing there is a yield sign in front of me.

What of the second arrow? It is pointing from content in the causal order to the conceptual order. Sellars denies that sensings can ground the non-inferential beliefs (as reasons);

Two Dogmas pragmatism

The totality of our so-called knowledge or beliefs, from the most casual matters of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man- made fabric which impinges on experience only along the edges. Or, to change the figure, total science is like a field of force whose boundary conditions are experience. A conflict with experience at the periphery occasions readjustments in the interior of the field. Truth values have to be redistributed over some of our statements.