David Corfield
Friedman and DTT

Idea

(NB. DTT is dependent type theory.)

Could one see the difference between Quine’s and Friedman's accounts of science as basically the same as that between Russell and a dependent type theorist on language? Recall the contrast between interpreting

The present king of France is bald,

either as

There exists something which is the present king of France, anything which is the present king of France is the same as that thing, and that thing is bald,

or the dependent type theoretic way

France is a country, the present is a time, ‘King of cc at time tt’ for cc a country and tt a time is a dependent type, ‘present King of France’ is contractible (i.e., there’s precisely one), allowing the formation of the type ‘The present king of France’, for which assert ‘who is bald’ (via dependent sum).

This marks a difference between presuppositions being treated as parts of the whole statements in one large conjunction, rather than as that without which other things can’t even be expressed. So “The present king of France is bald” is false in the first case because one of the conjuncts is false, while in the second it is a proposition which cannot be formed as a part of the context is an incorrect judgement.

The former fits with Quine’s outlook that theoretical predictions are consequences of large conjunctions representing mathematics + the scientific theory + particular experimental conditions. When such a prediction confronts experience, if it fails, by modus tollens we have to make a choice of which part of the conjunction to give up, the so-called ‘Duhem-Quine problem’. All things being equal, we tend to modify peripheral beliefs sooner than deeply entrenched mathematical rules.

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. Reevaluation of some statements entails reevaluation of others - the logical laws being in turn simply certain further statements of the system, certain further elements of the field. (W.V. Quine, ‘Two Dogmas of Empiricism’, From a Logical Point of View (New York, 1963), p. 42.)

Friedman’s thought is that this gives an incorrect picture of how science changes. His is a hierarchical picture, where without one layer it would be impossible even to express the next.

It follows that without the Newtonian laws of motion Newton’s theory of gravitation would not even make empirical sense, let alone give a correct account of the empirical phenomena: in the absence of these laws we would simply have no idea what the relevant frame of reference might be in relation to which the universal accelerations due to gravity are defined. Once again, Newton’s mechanics and gravitational physics are not happily viewed as symmetrically functioning elements of a larger conjunction: the former is rather a necessary part of the language or conceptual framework within which alone the latter makes empirical sense. (Friedman 2002, pp. 178–9)

(Kant, Kuhn, and the Rationality of Science,’Philosophy of Science 69(2): 171–90.)

Tasks

  1. Type theoretic rendition of an example of Friedman’s from mathematical physics
  2. Friedmannian rendition of examples where type theory captures presuppositions.

Type theoretic rendition of an example of Friedman’s from mathematical physics

Consider this example from Friedman:

Newton’s second law of motion says that force equals mass times acceleration, where acceleration is the instantaneous rate of change of velocity (itself the instantaneous rate of change of position). So without the mathematics of the calculus this second law of motion could not even be formulated or written down, let alone function to describe empirical phenomena. The combination of calculus plus the laws of motion is not happily viewed, therefore, as a conjunction of propositions symmetrically contributing to a single total result: the mathematical part of Newton’s theory rather supplies elements of the language or conceptual framework, we might say, within which the rest of the theory is then formulated. And an analogous (if also more subtle) point holds with respect to the relationship between Newton’s mechanics and gravitational physics. The law of universal gravitation says that there is a force of attraction, directly proportional to the product of the two masses and inversely proportional to the square of the distance between them, between any two pieces of matter in the universe—which therefore experience accelerations towards one another in accordance with this same law. But relative to what frame of reference are the accelerations in question defined? Since these accelerations are, by hypothesis, universal, no particular material body can be taken as actually at rest in this frame, and thus the motions in question are not motions relative to any particular material body. Newton himself understood these motions as defined relative to absolute space, but we now understand them as defined relative to an arbitrary inertial frame – where an inertial frame of reference is simply one in which the Newtonian laws of motion actually hold (the center of mass frame of the solar system, for example, is a very close approximation to such a frame). It follows that without the Newtonian laws of motion Newton’s theory of gravitation would not even make empirical sense, let alone give a correct account of the empirical phenomena: in the absence of these laws we would simply have no idea what the relevant frame of reference might be in relation to which the universal accelerations due to gravity are defined. Once again, Newton’s mechanics and gravitational physics are not happily viewed as symmetrically functioning elements of a larger conjunction: the former is rather a necessary part of the language or conceptual framework within which alone the latter makes empirical sense.(Kant, Kuhn, and the rationality of science, pp. 178-179)

One approach might be to take a modern reformulation of Newtonian physics (recall how Elie Cartan reframed Newton’s space and time as a four-dimensional manifold), then use the reformulation of modern physics in HoTT terms.

One problem with doing this is that we lose the historical understanding of the theory. Try instead to give a dependent type theoretic account of Friedman’s example as it was conceived at the time. Or is it possible that this can only lead to reconstructive work, unfaithful to the original?

See here.

The interesting idea that principles and propositions change their status requires us to understand them in terms of their historical status. We need to be sensitive to existing priority relations.

Don’t we see in \HoTT versions of physics that aspects of the mathematics are written into the type theory itself, e.g., the choice of ‘cohesion’, ‘infinitesimal cohesion’, etc.? What could one do with Newtonian physics?

But perhaps there’s some choice as to what is context and what theory From context:

Generally, a context is thought of as relative to some underlying logical theory. This underlying theory will contain most of the assumptions of what constitutes validity; the context of an assertion in this theory will then include only those extra assumptions that may be used by that assertion. On the other hand, one could also think of the entire base theory as part of the context.

Still there’s an order. We put cohesion into the type theory to make cohesive type theory, but it needs sufficient strength of type theory first.

What of Poincaré‘s choice between a simple geometry and complicated physics or a complicated physics and simpler geometry?

Friedmannian rendition of examples where type theory captures presuppositions

Secondly, we can show with the When did you leave off beating your wife example the idea of presuppositions allowing us to express something. Imitating Friedman:

It follows that without the concepts of personhood, of marriage, of action, of intention, etc., the idea of someone leaving off beating their wife would not make any sense, let alone give a correct account of the empirical phenomena: in the absence of these concepts we would simply have no idea what the relevant moral-legal-ontological frame of reference might be in relation to which the claimed cessation of violence has taken place. Once again, facts about personhood, intention and action are not happily viewed as symmetrically functioning elements of a larger conjunction: the former is rather a necessary part of the language or conceptual framework within which alone the latter makes empirical sense.

What does Duhem-Quine look like here?

How would we falsify a prediction from “the present king of France is bald”. Or “the present king of france has a good head of hair”. Had he hair we would predict that there would be a bill for royal shampoo. On not finding this, what do we give up? Perhaps we missed the bill, perhaps he’s bald, perhaps there is no present King of France. Perhaps there are no such things as kings or countries.

The difference between asserting and explaining

In answering the charge by admitting “I have left off beating my wife”, I commit to a range of things. What happens when I reply to “Why have you left off beating your wife?”? I must explain my decision by giving a plausible psychological account of the change of heart.

What then of the difference between asserting “The sun and the Earth exert an attractive force on each other, proportional to the product of their masses and inversely proportional to the square of the distance between them, and accelerate according to these forces” and explaining why this happens? Or “Why does the Earth travel in a near ellipse about the Sun?”.

Is there a difference in these cases between the theoretic posits making themselves heard more directly in the ordinary language case? “The Earth travels in a near ellipse about the Sun” doesn’t have any mention of forces, accelerations, etc.

Last revised on May 4, 2021 at 04:09:09. See the history of this page for a list of all contributions to it.