### Logical foundations for mathematics? The first-order vs. second-order ‘dichotomy’? (Part IV of 'Axiomatizations of arithmetic...')

(It took me much longer than I had anticipated to get back to this paper, but here is the final part of my paper on axiomatizations of arithmetic and the first-order/second-order divide. Part I is here; Part II is here; Part III is here. As always, comments are welcome!)

3.

**Logical foundations for mathematics? The first-order vs. second-order ‘dichotomy’?**

Given the (apparent) impossibility of
tackling the descriptive and deductive projects at once with one and the same
underlying logical system – what Tennant (2000) describes as ‘the impossibility
of monomathematics’ – what should we conclude about the general project of
using logic to investigate the foundations of mathematics? And what should we
conclude about the first-order vs. second-order divide? I will discuss each of
these two questions in turn.

If the picture sketched in the previous
sections is one of partial failure, it can equally well be seen as a picture of
partial success. Indeed, a number of first-order mathematical theories can be
made to be categorical with suitable second-order extensions (Read 1997). And
thus, as argued by Read, there is a sense in which the completeness project of
the early days of formal axiomatics

*has*been achieved (despite Gödel’s results), namely in the descriptive sense countenanced by Dedekind and others.
Moreover, categoricity failure must not be
viewed as a complete disaster, if one bears in mind Shapiro’s (1997) useful
distinction between algebraic and nonalgebraic theories:

Roughly, non-algebraic theories are theories which appear at first sight to be about a unique model: the intended model of the theory. We have seen examples of such theories: arithmetic, mathematical analysis… Algebraic theories, in contrast, do not carry a prima facie claim to be about a unique model. Examples are group theory, topology, graph theory… (Horsten 2012, section 4.2)

In this vein, proofs of (non-)categoricity
can be viewed as a means of classifying algebraic and non-algebraic theories
(Meadows 2013). This means that the descriptive (non-algebraic) project of
picking out a previously chosen mathematical structure and describing it in
logical terms has developed into the more general descriptive project of
studying theories and groups of theories not only insofar as they instantiate
unique structures (i.e. non-algebraic as well as algebraic versions of the
descriptive project).

On the deductive side, things may seem less
rosy at first sight. In a sense, first-order logic is not only descriptively
inadequate: it is also deductively inadequate, given the impossibility of a
deductively complete first-order theory of the natural numbers, and the fact
that first-order logic itself is undecidable (though complete). It does have a
better behaved underlying notion of logical consequence when compared to
second-order logic, but it still falls short of delivering the deductive power
that e.g. Frege or Hilbert would have hoped for. In short, first-order logic
might be described as being ‘neither here nor there’.

However, if one looks beyond the confines of
first-order or second-order logic, developments in automated theorem proving
suggest that the deductive use as described by Hintikka is still alive and
kicking. Sure enough, there is always the question of whether a given
mathematical theorem, formulated in ‘ordinary’ mathematical language, is
properly ‘translated’ into the language used by the theorem-proving program.
But automated theorem proving is in many senses a compelling instantiation of
Frege’s idea of putting chains of reasoning to test.

Recently, the new research program of
homotopy type-theory promises to bring in a whole new perspective to the
foundations of mathematics. In particular, its base logic, Martin-Löf’s
constructive type-theory, is known to enjoy very favorable computational
properties, and the focus on homotopy theory brings in a clear descriptive
component. It is too early to tell whether homotopy type-theory will indeed
change the terms of the game (as its proponents claim), but it does seem to
offer new prospects for the possibility of unifying the descriptive perspective
and the deductive perspective.

In sum, what we observe currently is not a
complete demise of the original descriptive and deductive projects of pioneers
such Frege and Dedekind, but rather a transformation of these projects into
more encompassing, more general projects.

As for the first-order vs. second-order
divide, it may be instructive to look in more detail into the idea of
second-order extensions of first-order theories, specifically with respect to
arithmetic. Some of these proposals can be described as ‘optimization projects’
that seek to incorporate the least amount of second-order vocabulary so as to
ensure categoricity, while producing a deductively well-behaved theory. In
other words, the goal of an optimal tradeoff between expressiveness and
tractability may not be entirely unreasonable after all.

One such example is the framework of ‘ancestral logic’ (Avron 2003, Cohen 2010). Smith (2008) argues on plausible
conceptual grounds that our basic intuitive grasp of arithmetic surely does not
require the whole second-order conceptual apparatus, but only the concept of
the

*ancestral of a relation*, or the idea of transitive closure under iterable operations (my parents had parents, who in turn had parents, who themselves had parents, and so on). Another way to arrive at a similar conclusion is to appreciate that what is needed to establish categoricity by extending a first-order theory is nothing more than the expressive power required to formulate the induction schema, or equivalently the last, second-order axiom in the Dedekind/Peano axiomatization (the one needed to exclude ‘alien intruders’). Here again, the concept of the ancestral of a relation is a plausible candidate (Smith 2008, section 3; Cohen 2010, section 5.3).
Extensions of first-order logic with the concept
of the ancestral yield a number of interesting systems (Smith 2008, section 4;
Cohen 2010, chapter 5). These systems, while not being fully axiomatizable
(Smith 2008, section 4), enjoy a number of favorable proof-theoretical
properties (Cohen 2010, chapter 5). Indeed, they are vastly ‘better behaved’
from a deductive point of view than full-blown second-order logic – and of
course, they are categorical.

Significant for our purposes is the status of
the notion of the ancestral, straddled between first-order and second-order
logic. Smith argues that the fact that this notion can be defined in
second-order terms does not necessarily mean that it is an essentially
higher-order notion:

In sum, the claim is that the
child who moves from a grasp of a relation to a grasp of the ancestral of that
relation need not thereby manifest an understanding of second-order
quantiﬁcation interpreted as quantiﬁcation over arbitrary sets. It seems, rather,
that she has attained a distinct conceptual level here, something whose grasp requires
going beyond a grasp of the fundamental logical constructions regimented in ﬁrst-order
logic, but which doesn’t takes as far as an understanding of full second-order quantiﬁcation.
(Smith 2008)

**4. Conclusions**

My starting point was the observation that
first-order Peano Arithmetic is non-categorical but deductively well-behaved,
while second-order Peano Arithmetic is categorical but deductively ill-behaved.
I then turned to Hintikka’s distinction between descriptive and deductive
approaches for the foundations of mathematics. Both approaches were represented
in the early days of formal axiomatics at the end of the 19

^{th}century, but the descriptive approach was undoubtedly the predominant one; Frege was then the sole representative of the deductive approach.
Given the (apparent?) impossibility of
combining both approaches in virtue of the orthogonal desiderata of
expressiveness and tractability, one might conclude (as Tennant (2000) seems to
argue) that the project of providing logical foundations for mathematics itself
is misguided from the start. But I have argued that a story of partial failure
is also a story of partial success, and that both projects (descriptive and
deductive) remain fruitful and vibrant. I have also argued that an
investigation of the conceptual foundations of arithmetic seems to suggest that
the first-order vs. second-order dichotomy is in fact too coarse, as some key
concepts (such as the concept of the ancestral of a relation) seem to inhabit a
‘limbo’ between the two realms.

One of the main conclusions I wish to draw
from these observations is that there is no such thing as a unique project for
the foundations of mathematics. Here we focused on two distinct projects,
descriptive and deductive, but there may well be others. While it may seem that
these two perspectives are incompatible, there is both the possibility of
‘optimization projects’, i.e. the search for the best trade-off between
expressive and deductive power (e.g. ancestral arithmetic), and the possibility
that an entirely new approach (maybe homotopy type-theory?) may even dissolve
the apparent impossibility of fully engaging in both projects at once. It is
perhaps due to an excessive focus on the first-order vs. second-order divide
that we came to think that the two projects are incompatible.

At any rate, the choice of formalism/logical framework
will depend on the exact goals of the formalization/axiomatization. Here, the
focus has been on the expressiveness-tractability axis, but there may well be other
relevant parameters. Now, if we acknowledge that there may be more than one
legitimate theoretical goal when approaching mathematics with logical tools
(and here we discussed two, prima facie equally legitimate approaches: descriptive and deductive), then there is no reason why there should be a unique, most
appropriate logical framework for the foundations of mathematics. The picture
that emerges is of a multifaceted, pluralistic enterprise, not of a uniquely
defined project, and thus one allowing for multiple, equally legitimate
perspectives and underlying theoretical frameworks. A plurality of goals suggests a form of logical pluralism, and thus, perhaps there is no real ‘dispute’ between first-order and second-order logic in this
domain.

I really enjoyed this paper, I had not looked at these issues in this way before. My one small criticism is that you could maybe spend a bit more time on ancestral arithmetic and HoTT in relation to the deductive/descriptive `divide,' especially as paper will appeal/ be accessible to people without a lot of background in the formal foundations of mathematics.

ReplyDeleteThanks for your comment! I agree that as it stands, the paper basically presupposes that the reader knows more or less what ancestral arithmetic and HoTT are all about. My hesitation is that if I start elaborating further on them, then I'll have to tell a much longer story, and the balance of the paper might be disrupted. For ancestral arithmetic in any case, Smith's paper has everything I might want to say :)

DeleteI really enjoyed all four parts of this article! It is nice to see such a balanced account of the deductive/descriptive divide with a positive outlook.

ReplyDeleteI have also been following the development of HoTT, mostly from the sidelines, but with great interest in what this fresh point of view might bring to foundations of mathematics. It is interesting to see that the deductive/descriptive clash is happening within HoTT itself. On the one hand, some seem to support the view that homotopy type theory is the language of higher topos theory. On the other hand, some seem to support the view that homotopy type theory is a better way to understand mathematical reality. A few months ago, I asked on MathOverflow whether HoTT had an intended model and Andrej Bauer explained that points of view are mixed in the HoTT community: http://mathoverflow.net/a/136369 (See also this recent exchange between Voevodsky and Shulman on the HoTT newsgroup to see how the two perspectives interact.)

Interestingly, there appears to be a third point of view which is relatively unaffected by the divide. Some who arrived at HoTT from the type-theoretic side seem to focus on the computational interpretation of homotopy type theory. Rather than being interested in describing mathematical objects, proponents of this perspective are more interested in describing mathematical activity. I find this point of view very refreshing and I am very curious to see what kind of impact this perspective will have on the development of HoTT.

Thanks for the comments, Francois (somehow I'm not able to 'reply' to your comment directly). It's very interesting to hear that within the HoTT community itself there is some disagreement as to what the whole project is about, so to speak. I'd say that the third point of view which you describe is basically a 'deductive' point of view, in the sense that it aims at describing mathematical activity, as you put it. At any rate, while it is still too early to say, I do think that HoTT may fundamentally change how we think about the deductive/descriptive divide, even if there are differences in emphasis among the HoTT community itself.

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