© John Woods. Informal Logic, Vol. 33, No. 2 (2013), pp. 292-331. 

Epistemology Mathematicized 

 
JOHN WOODS     
 
University of British Columbia & University of Lethbridge 

Department of Philosophy 

University of British Columbia 

1866 Main Mall E370 

Vancouver, BC 

Canada V6T 1Z1 

john.woods@ubc.ca 

 
Abstract: Epistemology and informal 

logic have overlapping and broadly 

similar subject matters. A principle of 

methodological symmetry is: philo-

sophical theories of sufficiently simi-

lar subject matters should engage sim-

ilar methods. Suppose the best way to 

do epistemology is in highly formal-

ized ways, with a large role for math-

ematical methods. The symmetry 

principle suggests this is also the best 

way to do the logic of the reasoning 

and argument, the subject matter of 

informal logic. A capitulation to 

mathematics is inimical to informal 

logicians, yet formal methods and 

mathematical models are an emerging 

force in epistemology. What is to be 

done? What’s sauce for the goose of 

epistemology is sauce for the gander 

of informal logic.  

 

 

Résumé: L’épistémologie et la logi-

que non formelle partagent des sujets 

à peu près similaires. Selon un princi-

pe de symétrie méthodologique, des 

théories philosophiques de sujets suf-

fisamment similaires devraient em-

ployer des méthodes similaires. Sup-

posons que la meilleure façon de faire 

l'épistémologie est extrêmement for-

malisée, et a un rôle important pour 

les méthodes mathématiques. Le prin-

cipe de symétrie suggère que c'est 

aussi le meilleur moyen de faire la 

logique du raisonnement et de l'argu-

mentation, ce qui est l'objet de la 

logique informelle. Une capitulation 

aux mathématiques est hostile aux 

logiciens non formels, mais les mé-

thodes formelles et les modèles 

mathématiques sont une force émer-

gente dans l'épistémologie. Que faut-il 

faire? Ce qui est bon pour l'oie de 

l'épistémologie est bon pour l’autre de 

la logique non formelle.   

Keywords: Bayesianism, conceptual adequacy, formal argumentation, formal 

epistemology, informal logic, mathematical, normative arithmetic, symmetry 

principle 

 



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293 

On [the] traditional view of the subject, the phrase ‘formal 

logic’ is pleonasm and ‘informal logic’ oxymoron. 

      John Burgess 

 

1.  Keeping mathematics in its place 

 

Notwithstanding their growing influence in epistemology and their 

established presence in probability theory and decision theory, 

mathematical methods haven't had much play in mainstream in-

formal logic.
1
 It is true that informal logic (IL) theorists tend to 

concede to modern formal logic a role in characterizing the notion 

of formal validity, and to the probability calculus a role in charac-

terizing the relation of inductive strength. But by and large these 

developments are borrowed from the areas of enquiry in which 

they first arose, and informal logicians don't often involve them-

selves in the further theoretical advancement of those disciplines. A 

smallish exception is the use of computational methods to model 

dialogical argument.
2
 Even so, most IL people who approach their 

work dialogically ignore or reject the more aggressively mathemat-

ical character of formal work in logics of dialogues and games.
3
 

 Informal logicians have met with success in characterizing 

modes of reason and argument which instantiate structures not 

well-caught (or catcheable at all) even in the run of the mill formal-

isms to which the IL community pays lip-service. A good example 

of this are contributions of Trudy Govier and others to the logic of 

                                                 
1
 “Mainstream informal logic” is a flexible expository convenience. As used here 

it signifies literatures of the sort generated by the journals of record, Informal 

Logic and Argumentation, and the monograph series Argumentation Library 

(Springer) and Studies in Logic and Argumentation (College Publications), as 

well as the numerous Proceedings of ISSA Conferences (Sic Sat). See also John-

son (1996). 
2
 See, for example, Reed and Norman (2003) and, more recently, the journal 

Argument and Computation. 
3
 See, for example, Walton and Krabbe (1995) and Walton (1998). An important 

exception is Barth and Krabbe (1982). More extreme recent exceptions—

certainly exceeding the reach of “informal logic”—are Barringer et al. (2012a) 

and (2012b). For something more accessible, but robustly technical, see Wein-

stein (2013). 



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294 

conductive argument.
4
 So far as I know, no one in the IL camp has 

thought to approach this logic by way of mathematical models. It is 

not hard to see why. One of the more dominant IL ideas is that 

mathematics doesn't pay the freight in these branches of enquiry. If 

we search the IL literature we aren't likely to find much, if any-

thing, in the way of systematic demonstration of this sentiment. An 

indifference to mathematical entanglements, if not an outright hos-

tility to them, is IL’s default position. Still, perhaps the time has 

come for a bit more aggression. If mathematical models are not the 

way to go in the logics of everyday reasoning and argument, 

wouldn’t there be some value in actually showing this to be so? 

 Of course, “paying the freight” is a metaphor. In its weakest 

sense, mathematics pays (some of) the freight in a theory when it 

has the ancillary role of expository convenience. At more extreme 

levels, there will be a degree to which non-mathematical content is 

infiltrated by the theory’s strictly mathematical requirements. In 

which case, not only does mathematical modelling “pay the 

freight.” To a non-trivial degree it also “calls the shots.” 

 There is substantial though not universal agreement among 

informal logicians that the mechanisms driving the human animal’s 

behaviour as a reasoning and case-making being are of the same 

general family as those that serve his further cognitive ends—

decision making, planning, belief formation, belief change, infer-

ence, and so on. So seen, since reasoning and case-making are 

themselves broadly epistemic enterprises, their respective theories 

should reflect this kinship in correspondingly appropriate ways.
5
 

We might say at a minimum that, except for cause, subject matters 

of like type require theories of like type. This is the Symmetry 

                                                 
4
 See here Govier (1980) and Blair and Johnson (2012).  See also Hitchcock 

(2013). 
5
 For further discussion, see Johnson (2007). See also Siegel and Biro (1997), 

Biro and Siegel (2006) and (2011), Freeman (2005), Finocchiaro (2005) and 

(2013), and Woods (2013a). I don’t want to overlook the frequency with which 

both argumental and reasoning practices are directed to non-cognitive ends. But 

they are practices which nonetheless deeply implicate the procedures involved in 

the advancement of those agendas. 



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295 

Principle. Since the correspondences are loose rather than one-to-

one tight, the Principle in turn is more approximate than strict. But 

I would say that it give symmetry enough to be getting on with. 

 This would be the right place to take particular note of rising 

tide in the theory of knowledge. It is the progressive entrenchment 

of formal epistemology as an established rival of analytic and natu-

ralistic approaches.
6
 Of the three, perhaps it is formal epistemology 

that hasn’t quite made it into the mainstream. But its prospects for 

promotion are far from bad; indeed they are brimming.
7
 The formal 

epistemologies I want to consider make a serious claim to mathe-

matical robustness. They are epistemologies in which mathematics 

pays a good deal of the freight and calls some of the important 

shots.  

 Perhaps there are some informal logicians who are ready to 

accept the prospects of formal encroachment with a certain equa-

nimity. But I hardly think that, if any at all, these brave souls exist 

in any numbers to speak of. This leaves the majority with three op-

tions. One is to overturn the Symmetry Principle. Another is to dis-

arm formal argumentation theory head-on. The third is to retain the 

Symmetry Principle and to deal with formal epistemology heads 

on. Of the three, the third strikes me as the best bet. The first is im-

plausible on its face. The second is discouraged by the fact that the 

wholesale invasion of informal logic by aggressively mathematical 

methods of argument has barely begun to happen. In contrast, for-

                                                 
6
 See, for example Gärdenfors (1988), Hendricks (2005) and Shoham and Ley-

ton-Brown (2009). Except for one chapter, all of Hendricks (2005) is a survey of 

formal epistemologies, and the book’s large bibliography is almost entirely de-

voted to their literatures. On the other side, modern analytic epistemology is in 

the abundant slipstream of the conceptual analyses favoured by Moore and Rus-

sell. Modern developments in naturalized epistemology, also by now well-

established, ensue from Quine (1969). 
7
 See again the journal Argument and Computation, especially, the guest editor’s 

note of volumes 2-3 of 2012 (van Benthem, 2012a, 2012b). See also the “formal 

models” sections of Gabbay and Woods (2003a) and (2005). A standard  formal-

ist textbook is Besnard and Hunter (2008). It is worlds away from Govier (2013), 

Johnson and Blair (2006) and, yes, from Woods, Irvine and Walton (2004) as 

well. 



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296 

mal epistemology now lays fair claim to a permanent seat at the 

councils of mainstream theory of knowledge. It offers us a bigger 

target, more settled and more easily hit.
8
  

 

 

2.  Modelling human behaviour 

 

Here now are two different approaches to the role of formal models 

in philosophical enquiry at large, not just epistemology. 

 
Now that we all understand the virtues of a model-theoretic 

semantics satisfying general Montagovian standards of rig-

our and clarity, there is joy in playing around with virtually 

every specific detail of Montague’s original paradigm. The 

following  illustrate various aspects of this new wave of 

free speculation…. 

 Johan van Benthem (van Benthem 1979), p. 337) 

 

There may be principles so central that anyone who rejects 

them and nonetheless uses the usual name for the concept 

may be said not to be using the usual concept but another of 

the same name. Such principles may perhaps be called “con-

ceptual,” and by those who accept them as truths, “concep-

tual truths.” We may perhaps be hypothetically obliged to 

accept them as true if we accept the concept. But we are not 

categorically obliged to accept them as true, since we are not 

obliged to accept the concept. We may even on the contrary 

be obliged to reject the concept . 

   John Burgess (Burgess 1998, p. 142) 

   

I want to examine the tensions exhibited in this pair of remarks. 

For ease of future reference we might call these the “Benthemite” 

                                                 
8
 Naturalized epistemology has made the “big time” in a way that formal episte-

mology has yet to do. Naturalized epistemologists routinely crop up in the phi-

losophy curricula of our universities. Formal epistemologists sometimes do, but 

not routinely; not in philosophy departments. This, I think, is subject to change, 

sooner rather than later. 



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297 

and “Burgessian” orientations and the tension between them the 

pull between a theory’s “conceptual adequacy” and its “technical 

or formal virtuosity.”
9
   

 Model based reasoning has long been a staple of theory-

building in the physical, biological and social sciences. Its presence 

in the philosophy of science is a more recent development.
10

 More 

so its influence on epistemology. When a model based theory has 

an empirical subject matter—whether the dynamics of natural se-

lection in populations, or the slipperiness of the freshly Zambonied 

ice at Madison Square Garden, or capital flows in deregulated mar-

kets—there is a deeply structured semantic fact about its models. It 

is that in some essential way what holds in the models fails to hold 

on the ground. Sometimes a theory of this kind will have a further 

feature which is said to explain this structural gap. When the sub-

ject matter of a theory is a human practice of a given kind K, it is a 

theory of K-behaviour or a K-theory. When such a theory advances 

assessment standards it is a normative K-theory. For generality 

let’s speak of N-theories. There are, of course, more N-theories 
than you can shake a stick at. To keep things manageable, I pro-

pose here to restrict their range to those that meet the following two 

conditions. The classes of K-event they investigate are of a broadly 

epistemic kind, and the methods of investigation, in a sense that 

will be further explained as we go along, are mathematical. So we 

should think of our reflections as a general sort of commentary on 

formal epistemology, especially its Bayesian variations.
11

 To that 

end, I want to tarry awhile with the three questions: 

 

                                                 
9
 To be clear: The historical Burgess is not himself a radically inflexible Bur-

gessian. The same is true for van Benthem’s own Benthemitism. 
10

 The use of models in mathematics and logic has a more settled and stateable 

history. See, for example, Badesa (2004) and Hodges (2007). As far as I know, 

comparable survey works for models in science have yet to be written. 
11

 Here, too, expository convenience calls for a certain generalized latitude. 

There is scarcely a principle of any importance that Bayesians in the flesh don’t 

disagree about. 



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298 

1. To what extent is it incumbent on a N to heed the ins-and-
outs of K-behaviour on the ground, and how is this done? 

2. How should a N of K-behaviour balance (if at all) analytical 
elucidation of its target concept—the concept of K—with the 

requirement that N be a mathematically well-made theory? 
3. What is the source of N’s normative legitimacy, and how 

does the N-theorist have access to it? 
 

 

3.  Concerning question (1)  

 

Question (1) asks for the extent to which it is incumbent on a theo-

ry of the assessable K-practices of human beings to heed the ins-

and-outs of K-events as they actually occur. Among philosophers 

of broadly analytic persuasion there is sizeable support for the idea 

that part of the answer has to do with the recognizability of a theo-

ry’s subject matter. Even at high levels of mathematical complexi-

ty, a good many N-theories are influenced by a simple principle of 
Burgessian import: 

 

CONCEPTUAL RECOGNIZABILITY: Whatever its other 

features, a N-theory of K should try to make the concept of 
K-events or K-practice recognizably present in its theorems. 

 

So constrained, a mathematical theory of a subject matter K which, 

except for the theory’s tutelage no one in his right mind would rec-

ognize as an account of K-behaviour, would be a K-theory in name 

only.  

 Most logicians will be aware of a dispute between dialethic 

and non-dialethic approaches to true contradictions, with Quine 

making an unrecognizability charge and Priest (among others) re-

sisting it.
12

 Quine’s complaint was that in allowing for the truth of 

                                                 
12

 A small grammatical point: The coiners of the term spell it “dialetheic” (simi-

larly “dialetheism”). Nothing in the grammar of either Greek or English justifies 



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299 

certain select statements of the form 
⌐
A  ~A

¬
, dialetheists had 

made the concept of negation unrecognizable. Quine’s complaint is 

an especially strong one. He thinks that even with the tutelage of-

fered up by dialethic logicians, any allowance for a true contradic-

tion obliterates the very idea of “not.” Without trying to settle that 

matter here, there is value in mentioning it. It brings to the fore the 

conceptual recognizability condition on theories.
13

 It invites us to 

ask whether, all things considered, this is a plausible constraint. It 

bids us consider how Burgessian we should want to be.  

 

 

4.  Concerning question (2) 

 

Question (2) is tightly connected to the issues raised by question 

(1). It asks whether a solid basis exists for determining a correct 

balance between a theory’s mathematical virtuosity and its concep-

tual clarity. Consider a N that gets the facts on the ground wrong, 
that is, wrong on purpose. Suppose again that N is a logic of prem-
iss-conclusion reasoning or rational decision-making. Suppose that 

it decrees that a rational agent is one who closes his beliefs under 

consequence and takes his decisions on the basis of perfect infor-

mation. Of course, these are stipulations massively defected from 

in actual practice. If we persisted with them, we would encumber 

ourselves with a passing strange notion of rationality. If our logic 

allowed that an agent is rational to the extent that he revises his be-

                                                                                                              
the second occurrence of ‘e.’ “Aletheia” gives us “alethic.” Why wouldn’t “di-

aletheia” give us “dialethic”? 
13

 For Quine’s complaint see Quine (1970/1986), p. 81, and for a reply see Priest 

(1999). A further recognizability dispute arises from the Routley star semantics 

for first degree entailment (FDE). Here too negation is the issue. Star worlds lack 

an intuitive characterization. Every world has a unique star world, and the star 

world of the star world of a given world is that same given world. In the event 

that a world has itself as its star world, negation has a classical structure. Other-

wise, a sentence 
⌐
~ A

¬
  is true in w just in case A is false, not in w, but rather in 

its star world w*. Informally, 
⌐
~A

¬
 is true in this world if and only if A’s falsity 

is possible here. For a discussion of the unrecognizability wrinkles occasioned 

by this treatment of “not,” see Restall (1999) and Woods (2012). 



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300 

liefs and makes his decisions in accordance with this logic, then its 

take on human rationality could only be that rationality is precisely 

what human beings lack under precisely those circumstances in 

which the rest of us in our untutored moments would think it not 

lacking at all. Many of us, philosophers and others alike, would be 

of the view that any logic of reasoning and decision-making that 

made all of us sweepingly and systematically irrational is a logic 

guilty of distortion on a scale that qualifies as conceptual misanaly-

sis. In shorter words, any such logic would sanction theorems 

whose falsity is a matter of conceptual necessity. They are theo-

rems that a Burgessian would have us reject. 

 I have asked whether the K-recognizability condition is a 

plausible requirement for N-theories. The answer presently at hand 
is in the affirmative: The theorems that make K-hood unrecogniza-

ble are false as a matter of conceptual necessity. A doubter might 

demand to know why this would matter? The answer is that it 

would matter if a certain very influential appreciation of normative 

K-theories were true. It would matter if a theory’s role were to pro-

vide a conceptual analysis of K. If so, 

 

CONCEPTUAL MISANALYSIS: N-theories that make their 
target Ks unrecognizable misanalyze those Ks and, in so do-

ing, fail the conceptual adequacy condition on theories.  

 

COROLLARY: Given a Burgessian orientation, such theo-

ries should be rejected. 

 

The conceptual recognizability constraint doesn’t lack for critics. 

Those who like it tend to give to “our” pretheoretic intuitions a 

load-bearing role in philosophy. Experimental philosophers aren’t 

so sure, and the same can be said for rational reconstructionists of 

reductive stripe.
14

 Then, too, there are philosophers who demand 

                                                 
14

 See in the first instance Knobe and Nicols (2008), and in the second Carnap 

(1928/1967). 



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301 

recognizability in one kind of intellectual context and yet are quite 

happy to do without it otherwise.
15

 

 It is a fact of some importance that not every N with an em-
pirically instantiated subject matter whose theorems are false on 

the ground makes its subject matter unrecognizable. Some such 

theories have theorems that are approximately true to what actually 

happened, hence might plausibly be said to be theories that make 

their target properties somewhat recognizable or quasi-

recognizable. So the distinction between having and lacking empir-

ically false theorems does not map theories to the distinction be-

tween those failing and those meeting the condition of K-

recognizability. Even so, we should also be careful not to lose sight 

of the fact that not every empirically false theorem is approximated 

to thus. For example, as we’ve seen, the axiom by which belief is 

closed under consequence is infinitely unapproachable by the in-

ferences of real life. I’ll come back to this in section 12. 

 

 

5.  Empirical sensitivity 

 

Empirically instantiable model based theories are empirically sen-

sitive:  

 

EMPIRICAL SENSITIVITY: An empirically sensitive model 

based theory of an empirically instantiated subject matter is 

one that pays principled attention to empirical deviations 

from its theorems. Attention is principled when it includes a 

determination of whether or not the empirical discomport-

ments overturn the theory or require some amendment of it.  

 

Of course, the same is true of model based theories of all kinds, not 

just the normative ones. Accordingly, we might propose that a 

                                                 
15

 Quine again comes to mind. Quine demands it for logic. He rejects it for phys-

ics where among other things, “a body is now a class of quadruples of sets of 

numbers according to an arbitrarily adopted system of coordinates. We are left 

with just the ontology of pure set theory” (Quine, 1981, p. 17). 



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302 

good way for any model based theory with empirically false theo-

rems to preserve the recognizability of its subject matter is to ad-

vance those theorems with empirical sensitivity. It is more easily 

said then done. 

 There are two main forms that theory’s empirical sensitivity 

might take. The theory might recognize the data on the ground as 

having the force of counterexamples, and seek to amend itself ac-

cordingly. Or it might take the normative high ground and resolve 

the conflict by negative assessment of deviant behaviour on terra 

firma. Both these options are affixers of blame. The first blames 

the conflict on the theory. The second blames it on practitioners.  

 There are problems with these approaches. If the theory is a 

theory of assessable human behaviour, the first option of holding 

its feet to the fires of empirical fact raises difficulties of its own. 

One is that the restoration of empirical fidelity often-times erodes 

the theory’s formal elegance and derivational power. In plainer 

words, empirical fidelity may cost a theory some of its most prized 

theorems. (A somewhat Benthemite note.) So there is a tension be-

tween technical adequacy and empirical fidelity, to which any rap-

prochement between formal and informal approaches to such theo-

ries must pay heed. The second option of ganging up on noncom-

pliant practitioners for falling short of what N demands of them 
raises question (3), which will occupy us just below.  
 

   

6.  Concerning question (3) 

 

Question (3) returns us to the problem of normativity. It is directed 

to theories embodying normative presumptions about good and bad 

practice. Where, it asks, is the source of the theory’s normative au-

thority, and how does the theorist have access to it? This we might 

call the normative authentication problem. A commonplace exam-

ple of a normatively presumptuous theory is logic. It used to be 

widely assumed and still is, more’s the pity, that a piece of reason-

ing is discredited if it violates a logical law. Simple inspection re-

veals that most of what is ordinarily taken to be good reasoning on 



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303 

the ground is invalid in the logician’s sense.
16

  But, if the blame 

lies with the practitioner, not the theory, this raises a good and nec-

essary question. What, if not getting the facts on the ground wrong, 

would discredit a theory? A second question lies closely by. It is a 

question posed by the rampant pluralism that has overtaken logic.
17

 

If two logics disagree on a logical principle or a rule of inference, 

how, if at all, is the conflict resolved? A not uncommon answer 

involves the system relativities embodied in Carnap’s Principle of 

Tolerance:  

 
In Logic, there are no morals. Everyone is at liberty to build 

his own logic, i.e., his own form of language, as he wishes. 

(Carnap 1937, p. 52) 

 

Any one system is just as good a logic as any other, no matter how 

incompatible. There is no fact of the matter about whether this or 

that is a theorem or whether thus or so is a valid rule. Something is 

a law of logic only relative to the system that proclaims it so. 

  There is much to dislike about the system relativity manoeu-

vre.
18

 I will mention just one bit of it.
19

 The system relativity ap-

proach to a logic robs it of the normative authority to call the shots 

for practices on the ground.  If there were no fact of the matter 

about whether an argument is valid there could be no fact of the 

matter as to whether an argument on the ground is good or bad by 

way of invalidity. It is true that logicians of relativist inclination are 

favourably disposed toward relativized normativities, reflected in 

judgements of the form “That argument is bad modulo the norms 

                                                 
16

 Even if, as is often the case, the validity requirement is relaxed to include the 

inductive strength requirement, most of what is taken to be good reasoning 

doesn’t meet it either. The exacting regulae of the statistico-experimental sci-

ences are in the general case too much for the individual human reasoner to han-

dle. See here Woods (2013). 
17

 Woods (2001), (2003), Beall and Restall (2006), Field (2009) and Woods 

(2011). 
18

 See, for example, Siegel (1987), a classic. 
19

 Other reservations about the theorists’ privileged access to normative authority 

are discussed in Gabbay and Woods (2003b).                                                     



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304 

of system S.” But as long as they remain system relativists about 
logic, there is no way that logic gives us to close the gap between 

“bad in S ” and “bad.” So either a judgement of “bad” is unsup-
ported by a supportable judgement of “bad in S,” or it is supporta-
ble, but not by logic. Either way, an argument’s goodness or bad-

ness cannot be determined by logic. It is determined, if at all, by 

the non-logical decision to adopt this logic rather than that one. 

The normative authentication problem is left untouched. 

 It is sometimes supposed that a theory’s K-norms are validat-

ed when they stand in a relation of reflective equilibrium to settled 

K-practice. According to some N-theorists, it is the sort of balance 
that is achieved when “the model may correct our intuitions and 

make them more precise” or “our intuitions may suggest a correc-

tion of a philosophical model” (Hartmann 2009, p. 23). The critical 

factor here is the scope of “our.” If the “our” of our intuitions is all 

of us, then the last thing that can be said of the idealizations of 

formal epistemology is that they are in reflective equilibrium with 

our cognitive practices. On the other hand, if the “our” of our intui-

tions is some or other self-selected group of formal epistemolo-

gists, then the old question recurs: How in the world did it get to be 

the case that they, of all people, were the very ones to achieve 

command of these insights?   

 Howard Raiffa is a founder of rational decision theory (Raif-

fa 1968). George Kennan was the US State Department’s chief ar-

chitect of the Cold War doctrine of containment, one of whose pil-

lars was the Marshall Plan (Kennan 1985). The containment strate-

gy was a brilliant success and Kennan’s decisions were of far-

reaching importance. But Kennan’s decisions routinely fell short of 

the Raiffa norms. For one thing, in all his long life George Kennan 

never once closed his beliefs under consequence. Never in his long 

life did he get finitely close to perfect information. Who, in some 

apt children’s words, made Howard Raiffa king of the castle?
20

  

                                                 
20

 There is an amusing story about Raiffa, perhaps too good to be true, but told as 

true by Paul Thagard in a lecture given at Sestri Levante to the MBR-12 Confer-

ence on Model Based Reasoning in June 2012. When Raiffa was at Columbia he 



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305 

 Left undealt with the normative authentication problem for 

N-theories is a standing embarrassment and, some would say, a 
deal-breaker. 

 

 

7.  Transfinitely false idealizations 

 

Model based science is chock-a-block with devices that deliver de-

liberate and benignly intended distortions of what happens on the 

ground. Leading the list of distorters are idealizations, which assert 

as true what is false on the ground—think here of perfectly rigid 

rods and infinitely large populations—and abstractions, which sup-

press as unimportant what is true on the ground—think here of the 

scalene triangle. Some idealizations are limits to which the facts on 

ground approach in some finite degree. But some are not in the 

least approachable. In population genetics, populations are infinite-

ly large. The smallest actual population and the largest actual popu-

lation are equally far removed from the ideal, namely infinitely so. 

In neoclassical economics, utilities are infinitely divisible, leaving 

the smallest pleasures of real life infinitely less divided than the 

model’s ideals. The same again is true of the closure law for be-

liefs. If beliefs were indeed closed under consequence, then the 

cardinality of the conclusions an actual agent infers from anything 

he believes must fall infinitely short of the cardinality of the con-

clusions the ideal agent draws.
21

 Idealizations of this sort are not 

                                                                                                              
received an offer from Berkeley and was having a difficult time making up his 

mind. Having become aware of this quandary, Raiffa’s colleague Ernest Nagel 

proposed that the relevant facts be plugged into Raiffa’s own decision-model, 

the button pressed, and the right answer patiently awaited. “Come on, Nagel,” 

the agitated Raiffa replied, “this is serious!” 
21

 Perhaps it will be suggested that a finitely unattainable limit sanctions a notion 

of finitely attainable greater and lesser degrees of approachability. Thus, we 

might find ourselves wanting to say that of the consequences you draw from A 

outnumber the consequences I draw from it, then, although neither of us approx-

imates the consequence-drawing in any finite degree, you  approach the limit 

more robustly than I. My view of this is that it doesn’t help much with the point 

at hand. If approachability in the present sense is allowed to displace approxima-



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306 

only not quite untrue empirically. They are utterly untrue empiri-

cally. They are as we said, infinitely unapproachable or, as we 

might also say, transfinitely false. They are propositions that ap-

proximate to truth in no finite degree.
22

 

 We come back to the point that a scientific theory that relies 

in any essential way on distortions such as these is expected to pay 

for them at the empirical checkout counter. But not all empirically 

distorting theories are capable of, much less designed for, such ne-

gotiations. If a theory is not designed for empirical payoff, and yet 

transfinitely distorts what happens on the ground, what justifies this 

cost? One answer of course, is its normative rather empirical pay-

off. But this runs us head-long into the as yet unsolved normative 

authentication problem. 

 Another response to the gap-problem is the theory’s technical 

virtuosity. It is an answer according to which a theory may get the 

facts on the ground wrong, it may lack a persuasive normative au-

thority, and it may be guilty of conceptual mismanagement, but by 

golly, it’s a lovely piece of mathematics! Which brings us back to 

question (2), and the Benthemite idea of “playing around” with 

“new wave[s] of free speculation.” 

 

 

8.  More about question (2) 

  

                                                                                                              
bility, it still remains that, for any given number of consequences drawn, it is 

always more rational to draw a greater number of them than that. While we’re on 

this point, much the same reservations can be directed to suboptimality theories 

which seek to reduce the severity of maximizing ideals by replacing them with 

gentler alternatives. Various efforts to lighten up, in the one case by fuzzifying 

probability measures and in the other by relaxing the strictness of Bayesian 

standards are well-discussed in the literature. See in the first instance Jeffrey 

(1984), van Fraassen (1990), and in the second van Fraassen (1984), (1999). My 

view of the lightening up movement is twofold. It doesn’t go far enough, and it 

claims a normative legitimacy it fails to establish. 
22

 Transfinitely false models are discussed in Woods (2013b). See also Woods 

and Rosales (2010). 



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Here is an example of the contrast between a theory’s conceptual 

and mathematical adequacy. It is drawn from Bayesian epistemol-

ogy. Bayesians differentiate their approach from traditional episte-

mology, which proceeds by an examination of epistemic intuitions 

and the concepts embedded therein. Since the traditional episte-

mologist is typically a proponent of conceptual analysis, he will 

have some sympathy for the concept-recog-nizability requirement. 

Bayesians have a different motivation. In the words of two of its 

able practitioners, “Bayesian epistemology, on the other hand, 

draws much of its power from the mathematical machinery of 

probability theory” (Hartmann and Sprenger 2011, p. 619). This 

approach to epistemology 

 
starts with mathematical intuition. The construction of 

Bayesian models is much triggered by what is mathematical-

ly elegant and feasible (e.g. Spirets et al. 2001). The mathe-

matics develops a life of its own (to adopt a phrase due to 

Hacking), and the comparison with intuitive examples 

comes only after the Bayesian account is given (Hartmann 

and Sprenger 2011, p. 619; emphases in the original). . 

Indeed it is easy to distort the problem under consideration 

by making implausible assumptions that make the modeling 

easier (Bovens and Hartmann 2003, p. 130).  

 

We have arrived at a critical juncture. I said earlier that the mathe-

maticization of a non-mathematical subject matter counts as ex-

treme to the extent to which its mathematical provisions infiltrate 

that subject matter. Bayesianism, as here described, is mathemati-

cized epistemology of high order. 

 Of course Bayesians don’t think of themselves as wholly in-

different to the conceptual implausibilities occasioned by their 

models. Theirs is not the Anything Goes line in epistemology. In 

fact, 

 

 it is often all too easy to get carried away by the formu-

lae. Not every partial derivative is meaningful—no matter 

how pretty the result. (Bovens and Hartmann 2003, p. 130) 



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However, as Hartmann and Sprenger also insist, the “Bayesian 

framework is very convenient for” the study of the intuitive data, 

owing in no small measure to the fact, as they suppose, that “it 

comes with principles that can be made plausible on rational 

grounds (Dutch Book arguments, Principal Principle, Bayesian 

Conditionalization)” (Hartmann and Sprenger 2011, p. 619; em-

phasis in the original).
23

 

 Note that this attention to “intuitive data” comes after the 

theory’s mathematical intuitions are catered for. Once the theory is 

up and running, the intuitive data can be brought to heel by the 

normative authority the mathematics is presumed to possess. Ac-

cordingly 

 

SHEDDING INTUITIONS: The mathematical theory owes 

no ultimate allegiance to intuitive concepts and is fully em-

powered to override those intuitions when they deviate from 

the theory’s provisions. 

 

If anything is true of belief change on the ground it is the sheer ex-

tent to which it deviates from the axioms of the probability calcu-

lus, making of these three elements—the Dutch Book theorem, the 

Principal Principle and Conditionalization—largely an academic 

matter. Strict Bayesians will not allow that these empirical devia-

tions in any way discredit the principles. For they are mandated and 

made impervious to this kind of empirical discouragement “on ra-

tional grounds,” that is, by the normative authority to regulate the 

concept of rationality in this way. But this just cycles us back to 

where we were only paragraphs ago, where it was proposed that the 

                                                 
23

 Informally, the Dutch Book theorem holds that it is irrational to shape the de-

grees of one’s beliefs in ways that deviate from the axioms of the probability 

calculus. The Principal Principle holds that if an agent knows the objective prob-

ability of a proposition A to be p and has no information to the contrary, then his 

rational degrees of belief in A must also be equal to p. Bayesian Conditionaliza-

tion asserts that the rational degree of belief in a proposition A after learning E is 

the conditional probability of A given E. 



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approved way to go is simply to calumniate standard practice for 

its deviations from the theory’s normative presumptions. 

 My own view is that Bayesians—indeed virtually all the go-

ing formalizers of human cognitive practice—are confused about 

the factors that confer normative authority on their theories. To ex-

plain this further, we would do well to give some thought to a pair 

of very old ideas in epistemology. One is that a normative theory 

cannot be upset by empirical facts. The other is that a mathematical 

theory is likewise impervious to empirical confutation.
24

 As we see 

from Hartmann’s and Sprenger’s remarks just now, Bayesians are 

drawn to an assumption to which, so far as I can tell, they have yet 

to give voice:  

 

MATHEMATICIZING NORMATIVITY: The central reason 

why N-theories can’t be overturned by recalcitrant experi-
ence is that since N-theories are mathematical theories, their 
normative authority derives from that same mathematical im-

perviousness to those discomportments.  

 

In so saying, we have a classical many-one confusion. Mathemati-

cal theories are impervious in this way. So are normative theories. 

Ergo, mathematics is normative. 

 We might be forgiven for thinking that Bayesians are pixilat-

ed by the power and majesty of mathematics. Arguably the greatest 

metaphysical discovery ever made by science is the extent to which 

the deep regularities of nature are mathematically expressible. We 

might also say that one of the greatest discoveries in the epistemol-

ogy of the natural sciences is that a mathematical theory of nature 

is capable of successful negotiation at the empirical checkout, 

hence capable of succeeding or failing on the strength of the obser-

vational predictions it manages to help generate. In the fullness of 

time, beginning with economists, social scientists set out to discern 

a similar link in the mental and behavioural sciences. Lately they 

                                                 
24

 Notwithstanding some significant reservations here and there.  See, for exam-

ple, Maddy (1997) and Bostock (2009). 



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have been joined by philosophers of a like tilt. It has not been a 

problem-free conversion.  

 Whole classes of mathematical theories of assessable human 

behaviour have risen up and flourished, undiscouraged by the 

recognition that, unlike physics, they cannot ground their findings 

or authenticate their assertions at the empirical checkout counter. 

Short of giving up, this leaves their proponents with a hybrid of the 

two alternatives that we’ve been examining. Normativize the theo-

ry, or mathematicize it. The hybrid marries the two in a particular 

way, in which, as we saw, normativization is achieved by mathe-

maticization. This I take to be the inclination of Bayesians of the 

Hartmann-Sprenger stripe. For how else are we to understand their 

assurances that when its pronouncements conflict with the intuitive 

data a theory driven by mathematical intuitions will make this a 

tolerable shortfall “on rational grounds”?   

 

 

9.  Rethinking mathematical intuitions 

 

Let me now pick up the threads of two sections ago. Ian Hacking’s 

metaphor in which a theory’s mathematics “takes on a life of its 

own” has an undeniable allure, and the extent to which it captures 

what actually happens in N-theories is absolutely striking. But it is 
not absolutely true. It is not true in ways that precludes a N-theorist 
questioning, case by case, his theory’s own provisions. This can be 

illustrated by a brief glance at the old problem of irrelevant con-

junctions. Suppose we have strong evidence that the Canuck’s are 

playing tonight. This should not also be evidence that the Canucks 

are playing tonight and the Leafs will win the Stanley Cup in 

2013.
25

 Yet without some resourceful tampering, this is precisely 

what the Bayesian treatment of evidence provides: 

 

                                                 
25

 Not everyone will know that the Canucks and the Leafs are National Hockey 

League teams and that the Stanley Cup goes annually to the NHL championship 

team. 



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Assume that E is evidence for A and that Pr(E|A . X) = 

Pr(E|A) for any sentence X consistent with A. Then E is also 

evidence for 
⌐
A  X

¬
, for any measure of support. (Fitelson 

2002)  

 

The irrelevance problem has spurred a spate of attempts to reshape 

the conditions on evidence, prompting one philosopher to remark: 

 
Thinking about the different measures of support suggests to 

me that there is no single clear cut question being asked 

when we ask “How much support does P get from G?” It 

would not be surprising if the same were true of the question 

“How much does evidence E support hypothesis A?” (Chris-

tensen 1999, p. 439)  

 

If time permitted, it might be interesting to review with care the 

main candidates—see here Crupi et al. (2007)—but that is not my 

purpose here. My purpose here is to ask  

 

Upon what grounds is it to be determined that a mathematical 

theory of belief change in which irrelevant conjunctions are 

derivable is bad mathematics?  

 

And with it a second question:  

 

If indeed such a theory is bad mathematics, what is it about 

the mathematics that makes it so?  

   

There are some easy but unhelpful answers we could consider. A 

mathematical theory is bad if it is inconsistent.
26

 A mathematical 

theory is bad if it is lazy—e.g., some of its key assertions aren’t 

proved. A mathematical theory is bad when it is fat—e.g., lots of 

its theorems are redundant. These we might think of as internal dif-

ficulties. Our interest here lies elsewhere. It concerns the external 

                                                 
26

 Or, with a nod to dialethists, it is promiscuously inconsistent. See Mortensen 

(1995). 



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312 

difficulty occasioned by landing one’s account of evidence in the 

lap of irrelevant conjunctions. Whence, then, the source of this 

badness? 

  We are thus moved to ask whether the mathematical theory 

of probability is mathematics. The answer, of course, is that it is 

not. We have but to touch on the core elements of Bayesianism to 

seek why this is so: 

 

1. Beliefs are had in real valued degrees. (Philosophy) 

2. Beliefs are subjective probabilities. (Philosophy) 

3. Probabilities obtain in real valued degrees. (Philosophy) 

4. “P” is a name-forming operation on sentences A. (Philoso-

phy) 

5. For all propositions A there is a real number x  [0, 1] such 

that P(A) = x. (Philosophy) 

6. P(A) = 0.4, and P(B) = 0.2, with A and B independent. 

(Assumed as fact) 

7. P(A  B) = P(A) x P(B) (Philosophy) 

8. 0.4 x 0.2 = .08 (Mathematics) 

9. P(A  B) = .08 (from 6, 7, 8) 

                                                        

Thus (9) arises from the five philosophical propositions that serve a 

foundational role, together with an assumed fact, two more philo-

sophical propositions and one rather measly calculation from the 

arithmetic of real numbers. For all the fuss that is made of the 

mathematical power of probability theory there is in a Bayesian 

appropriation of it for the purposes of epistemology hardly any-

thing that counts as mathematics. Even so, we should not conclude 

that mathematics doesn’t, after all, call the shots in Bayesian epis-

temology. Here is why. If you are one of those philosophers who 

think it important that a theory of knowledge recognize that belief 

comes in degrees, you may want to made degrees of beliefs calcu-

lably manipulable in your theory. If so, you would be well-advised 

to conceptualize belief in ways that allow this to happen. True, 

there is hardly any mathematics in Bayesian epistemology. But 

given the premium it puts on doxastic calculabilities it is hardly 



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avoidable that the arithmetic of the reals would call some of the 

theory’s non-mathematical shots. So we see that mathematicizing a 

non-mathematical concept can fall well short of making it a math-

ematical concept. A very little mathematics can call quite a big 

shot. 

 

 

10.  Where’s the mathematics? 

  

It bears repeating that the very idea that the concepts of epistemol-

ogy are mathematical concepts is risible on its face, and likewise 

the view that probability is mathematics. The statement that beliefs 

are probabilities is a philosopher’s claim, as is the statement that 

evidential support is conditionalization. The same is true of the as-

sertion that probabilities are real numbers and that probabilities 

obey the real-valued Kolmogorov axioms. Probabilities may be re-

al numbers, but nothing in analysis makes it so.  

 This should steel us against overstated expressions of math-

ematical dominance. It should get us to see that Hartmann and 

Sprenger have misspoken themselves in their 2011 entry in the 

Routledge Companion to Epistemology. A soberer appreciation of 

this dominance is to be found in other of Hartmann’s writings, no-

tably in “Modeling in philosophy of science,” which is part of a 

tribute to Patrick Suppes for his foundational work on modelling in 

the philosophy of science (Hartmann 2008).
27

 It is unnecessary to 

review this paper at any length to get out of it the points that matter 

for our present question. So I shall be selective in my pickings. 

 

NOT MATHEMATICS: Bayesianism is not a mathematical 

theory. It began its life as a philosophical theory of confirma-

tion. It is a philosophical theory that makes use of a mathe-

matical theory. The mathematical theory is a theory of de-

grees or amounts of confirmation. For example, the equation 

of evidential support with conditional probability is a philo-

                                                 
27

 See also Suppes (1962) and (2002). 



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sophical assertion. The equation of conditional probabilities 

with the right-hand side of Bayes’ theorem is a mathematical 

assertion.
28

 

 

DYNAMIC VARIATIONS: As its influence spread, Bayesi-

anism migrated to epistemology’s further precincts, and in 

due course coverage extended to belief change, decision-

making, and theory change, with a concomitant shift from 

synchronic to diachronic analyses—hence, belief change. 

These, too, are philosophical developments, and they too 

contain mathematical mechanisms for the regulation of such 

quantitative factors as enter into the operation of these dyna-

misms.
29

 

 

NORMATIVIVELY INTENTED: Contrary to the domi-

nance of mathematical intuitions thesis, Bayesianism’s philo-

sophical provisions are forwarded with epistemically norma-

tive intent, and virtually without independent justification. In 

particular, it is not expressly envisaged that they admit of 

mathematical authentication. Even so, that a set of philosoph-

ical propositions would admit of—indeed would welcome—

fruitful mathematical engagement is often take as philosophi-

                                                 
28

 “Bayesianism [is] a normative philosophical theory . Bayesianism formu-

lates constraints on the beliefs of a scientist” (p. 15). Bayesianism is also “a 

quantitative confirmation theory . While qualitative [philosophical] theories 

formulate criteria that inform us whether or not a piece of evidence E confirms a 

hypothesis H, quantitative [mathematical] theories of confirmation also tell us 

how much E confirms H (pp. 5-6). “But how are probabilities updated? . 

There is a lot of debate about this question amongst Bayesians and there is clear-

ly no consensus. Many, however, hold that the new probability measure should 

be the conditionalized old probability measure. I’ll not defend this choice, but 

will use it in the remainder [of this paper]. Note, however, that this choice is not 

dictated by mathematics. What is dictated by mathematics is how one evaluates 

the expression on the right side ” (pp. 5-6). Actually, as we saw, not even this 

is true. There is nothing in real analysis that tells us this. Bayes’ theorem is not a 

theorem of mathematics. It is a philosopher’s thesis. 
29

 See Bovens and Hartmann (2003), chapter 1 et passim. 



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cally encouraging. This is a point of significant equivocation 

in formal epistemology generally.
30

  

 

INTUITIONS REDUX: If pressed, however, Bayesians will 

concede that the normative authority of its pronouncements 

rests on our conceptual intuitions about the matters at hand, 

anyhow those of them that are currently in reflective equilib-

rium.  

 

Here is Hartmann himself on the last two of these points: 

 
Why should we believe in a philosophical model, accept its 

consequences and be content with an explanation that a 

philosophical model provides us with? After all, unlike sci-

ence, we cannot compare the model with “hard” data from 

experiments. All we have in philosophy is our intuitions, and 

all we can do is to aim at a reflexive [sic] equilibrium be-

tween our intuitions and the consequences of the model . 

Clearly, much more needs to be said about the assessment of 

philosophical models, but I have to leave this for another oc-

casion. (pp. 22-23) 

 

And here is a further word (from me) about reflective equilibrium. 

I have already observed that what standard cognitive practice is not 

in reflective equilibrium with are the normatively presumptive ide-

als of Bayesian epistemology. Still, I agree that our business-as-

usual cognitive practices probably are in reflective equilibrium 

with the actual norms, whatever they chance to be. 

 Let’s be clear, then. Bayesians don’t identify epistemic ob-

jects and properties with mathematical objects and properties. 

What they do is represent epistemic objects and properties as 

mathematical objects and properties. They do so, it is said, for the 

benefits that redound from this way of proceeding to the objects 

                                                 
30

 When neoclassical economics introduced the infinite divisibility idealization 

for utilities, it is easy to say why. It was done to enable economics to engage the 

mathematical fire-power of calculus. 



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and properties of epistemology. But what benefits? And how are 

they realized? 

  

 

11.  Representation 

 

The word ‘representation’ carries more meanings in philosophy 

than it can comfortably wear. There is a lifetime’s work, and then 

some, in bringing this idea to philosophical heel. My purpose in 

invoking it here is the occasion it gives briefly to consider one of 

the ways in which we might treat a well-known difficulty that at-

tends it. Representations are routinely called upon to remove the 

sting of false identifications. If someone says that beliefs are prob-

abilities, he has made a false identification. If he says that probabil-

ities are real numbers, he has made a false identification. If he says 

that person’s confidence in a belief are the odds he is prepared to 

accept in a contrary bet, that too is false. If someone says that for 

each real number in the unit interval there is a belief whose intensi-

ty it is, this is a further falsehood. Consider the case in which, hav-

ing himself advanced these claims, a theorist of Bayesian sympa-

thies is now prepared to grant their falsity. Suppose, even so, he 

remains drawn to these propositions, and is reluctant to give them 

up. Suppose he looks for a way of neutering their alienation from 

the facts of standard practice. He says: Yes, strictly speaking, be-

liefs aren’t probabilities (etc.), but it is nonetheless illuminating or 

instructive to represent these things as those things. It is illuminat-

ing and instructive because, in thinking of these things as those 

things, we attain a better appreciation of these (or other) things than 

would otherwise be so. 

 Of course, this is modelling talk of a kind that routinely at-

tends a scientist’s handling of natural phenomena. It fits especially 

well the notion of “phenomenological” model, typified by Bohr’s 

planetary model of the interior of the atom. Bohr’s contention was 

that thinking of quantum phenomena in that way was helpful in 

understanding the notion of quantum leap. Bohr was right. It did 

help. 



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  Representation is not FOB. There are excise duties to be 

paid. A representation of something would be of little moment if 

the way it got you to think of it made no contribution to your theo-

ry’s success. We have it by construction of the case that although 

electrons aren’t planets, it is illuminating to think of them that way. 

But this is unavailing if thinking of them in that way makes no ad-

vance at the checkout counter. Needless to say, no one thinks that 

that the planetary representation is a nonredundant premiss in the 

derivations that yield the observational predictions of quantum the-

ory. But I am prepared to concede—if only for the sake of argu-

ment and to speed things up—that Bohr’s representation may have 

played a causally efficacious role in the thinking up of the premiss-

es that do figure in those derivations.  

 In like manner, I am prepared to grant—with the same quali-

fications—that giving to the central concepts of epistemology 

Bayesian representations may in some distant future prove causally 

efficacious in finding new premisses for epistemology that some of 

us may come to like better than the ones we’ve got now. But what 

we shouldn’t concede is that the new premisses and the conclusions 

they link up with will vindicate the representations in the way that 

we are supposing for quantum theory. It is all down to the same old 

grief. Quantum theory has a future at the empirical checkout stand. 

Bayesian epistemology, on its own conception of itself, does not. 

This is a yawning difference. On present assumptions, the Bohr 

representation pays its taxes by virtue of the causal provocation it 

lends to the thinking up of a scientific theory that does well at the 

checkout counter. The Bayesian representations might be said to 

pay their taxes by virtue of the causal provocations they lend to the 

thinking up a theory of knowledge that Bayesians like better than 

the old one. That Bayesians take a liking to their own norms makes 

no dent on standard practice (not even their own). So the empirical 

deviation problem remains a problem unless Bayesians can find a 

more convincing way to solve the normative authentication prob-

lem. The normative legitimacy problem is left unsolved. Consid-

erations on an empirically accessible normativity are worked up in 



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318 

chapter 2 of my Errors of Reasoning: Naturalizing the Logic of 

Inference (Woods 2013a). Here, very briefly, is its gist. 

 In epistemology’s analytic/formalist orientations good prac-

tice is a matter of conformity to the requisite criteria of goodness. 

The criteria could be considered as rules/programs, and conformity 

would be seen as a kind of obedience/imple-mentation. The natu-

ralistic perspective is different. The goodness of practice is not a 

matter of obedience to rules or implementation of programs, whose 

purport is to define the meaning of “good practice (of that kind).” 

A piece of reasoning-behaviour is good when the causal mecha-

nisms which produced it are in apple-pie working-order and are 

operating here as they should. Thus reasoning well is more like 

breathing well than shaping and restraining one’s behaviour in di-

rect response to the Ten Commandments (whether from Heaven or 

Amsterdam). So far as is known empirically, the cognitive equip-

ment of beings like us is in good working-order and operating the 

way it is supposed to pretty much as a matter of course. Not al-

ways, to be sure, but enough of the time to ground a judgement of 

“normally.” In which case, we might want to seize on this and pro-

claim  

 

The NN-Convergence Thesis: In the absence of countervail-

ing considerations, in matters of human reasoning what nor-

mally happens is what should happen. In other words, the 

normative converges on the normal. 

 

 I present NN Convergence to give notice of it, but not  with 

any present expectation that it will quiet the normative anxieties of 

people with non-naturalistic leanings. It’s a gander-goose matter. I 

don’t like the nonnaturalist’s treatment of normativity. Very well, 

shouldn’t I be expected to do better? Shouldn’t, at least, I be pre-

pared to expose my own view to the critical eye of the other side? 

Yes, of course. But that question doesn’t have to be settled here. 

Mathematicized epistemology stands or falls independently of the 

fate of NN-Convergence. 

 



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12.  Mathematicization as synthesis 

 

When formal methods are applied to a concept, they constitute a 

formalization of it. When formal methods have a substantial math-

ematical character, they constitute a mathematicization of the con-

cept. When speaking of a concept’s meaning in pre-formalized lin-

guistic practice, let us say that we are speaking of its intuitive 

meaning or, equivalently, of the intuitive concept. Consider now 

our earlier question, “What is achieved by the mathematicization of 

a concept; for example, what do we learn about knowledge from a 

mathematicization of the concept of knowledge?” At the appropri-

ate level of generality, there are four different answers to this con-

cept-capture question.    

 

Analysis The mathematicization of concept K explicitizes the 

meaning it has in pre-formalization linguistic practice; that is, 

it captures K’s intuitive meaning.
31

 

 

Explication The mathematicization of a concept K preserves 

its pretheoretical meaning but does so in ways that gives to K 

a more aggressive clarity than it has in pretheoretical linguis-

tic practice. 

 

Rational reconstruction The mathematicization of a concept 

K involves an additional attribution to K of features not pre-

sent in pretheoretic linguistic usage, but in a manner that re-

tains enough of the intuitive concept to make it intelligible to 

say that its rational construction is a formalization of it, rather 

than some other. 

 

                                                 
31

 For lack of space, I pass over two important variations of the definitional con-

dition on formalization, namely, implicit definitions and contextually eliminating 

ones. For the first, think of the definition of the concept of number afforded by 

the axioms of number theory. For the second, think of the reduction of number 

theory to set theory attempted by Frege and Russell. 



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320 

Stipulation Here the mathematicization constitutes a nominal 

definition of a concept lacking a prior presence in pretheoret-

ical linguistic practice.  

 

The distinction between analysis and stipulation is roughly Kant’s 

contrast between analysis and synthesis. Analysis, says Kant, is the 

business of making concepts clear, and synthesis the business of 

making clear concepts. Analysis is the purview of philosophy, and 

synthesis the province of mathematics.
32

 In this taxonomy, explica-

tion and rational reconstruction are hybrids, with explication more 

analysis-like and rational reconstruction trending towards stipula-

tion. Similarly, analysis is the most Burgessian option, and stipula-

tion the most Benthemite. Explications and rational reconstructions 

are likewise hybrids, with explication tilting more to the Burgessi-

an and reconstructions more to the Benthemite. 

 There are, of course, grey areas at each of these borders. As 

Quine once quipped, one person’s explication is another’s stipula-

tion.  But there are some reasonably clear examples. Philosophers 

in the tradition of Moore think that the correct analysis of the con-

cept of material thing is inconsistent with (subjective) idealism. 

Mathematicians take the view that the axioms of plane geometry 

offer an explication of the intuitive concept of physical space. The 

notion of rational reconstruction we associate with Carnap. Its 

presence may be felt in Carnap’s attempt to formalize physical ob-

jects as logical constructions of sense data. Stipulation is the stock 

and trade of mathematics, as Kant noted. But it is also solidly at 

work in all of model based science. Populations are infinitely large; 

utilities are infinitely divisible; belief is closed under consequence; 

and deciders have perfect information. The analysis of a concept 

presupposes its intuitive presence in preanalytic practice. It re-

quires full recognizability. An analysis may be said to be conceptu-

ally faithful to the degree that it preserves the presence of its intui-

tive analysandum. At the other end of the spectrum, synthesis, or 

the stipulation of new concepts, places a premium on the clarity 

                                                 
32

 Kant (1764/1974) and (1800/1974). 



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321 

that can be got from an inventive mathematical virtuosity. As re-

gards the present concept engagement possibilities, it would appear 

that the formalizations effected by retrofitted logics fall oftener 

than not in the ambit of partial reconceptualization—hence of qua-

si-recognizability. It is well to emphasize, however, that option 

four—the most creative of them all—will in principle tolerate ele-

ments of the three prior grades of conceptual engagement.  

    

SHEDDING INTUITIONS (AGAIN): The single most dis-

tinctive feature of synthetic theories (as we may now call 

them) is their freedom in principle from any requirement of 

prior conceptual recognizability, and with it their liberation 

from the travails of conceptual analysis—full-blown or par-

tial, it doesn’t matter.
33

  

 

 While the idea of synthetic theories has its roots in Kant, it is 

to Carnap that we can turn for its instrumentalist justification. Any 

logic will do if it adequately fits our nonlogical purposes. A like 

permissiveness is advanced by Quine, though in mathematics, not 

logic, under the heading of legislative postulation. Quine’s nerv-

ousness about logic is anomalous, and in any event not catered for 

by his own pragmatism. Be that as it may, the idea of legislative 

postulation carries a lot of weight here. It provides that when one 

postulates legislatively, there are sentences that one makes true, 

and via the mechanisms of quantification, makes objects one makes 

to exist.  

 Perhaps the best known example of a stipulative theory with 

a happy ending is Riemannian geometry. No one, least of all its 

inventor, took Riemannian geometry as a description of the actual 

                                                 
33

 Thus a mathematical physicist wants to prove theorems about c* algebras in 

ever more general and robust settings. A theoretical physicist uses c* algebras 

for improved representations of physical systems that will eventually do well at 

the empirical checkout counter. Similarly, a mathematical biologist tracks gener-

alized solutions of Lotka-Volterra systems of idealized competitors. A theoreti-

cal biologist investigates biologically meaningful conditions for the coevolution-

ary stability of competition. 



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322 

space of the physical world. Its virtues were technical, notably its 

proof of relative consistency. It was a significant piece of mathe-

matical virtuosity but hardly up to the burdens of conceptual fideli-

ty. Riemann’s was not a semantic analysis of the concept space. 

But with the arrival of general relativity theory, it emerges that 

Riemann’s space is the empirically well confirmed description of 

big space.  

 There is no reason to think that Riemann himself anticipated 

this Einsteinean application of his abstract theory. Even so, non-

Euclidean geometry was ready to hand—on the shelf, so to 

speak—and was available for Einstein’s later appropriation. But 

this couldn’t have happened in the first place had it not been for the 

impressive technical virtuosity of Riemann’s achievement. People 

would not have paid attention to it otherwise.
34

  

 Modern logic makes indispensable use of mathematics. The 

methods of mathematics are embedded in the generative grammars 

of its formal languages and they drive its theories of proof and 

truth, and power the techniques of their linking metatheories. Log-

ic, of course, is not mathematics. The conjunction operator is unde-

fined for arithmetic. But logic has got some mathematics in it. It is 

not wrong to speak of modern logic as mathematical. We have in 

this a critical distinction. It is the distinction between subject mat-

ter and method. 

 Bayesian epistemology should be judged in the light of these 

contrasts. It is a mathematical epistemology in the sense in which 

mathematical physics is mathematical. It is a synthetic theory, also 

in the way that Kant and Quine say mathematics is. Bayesian epis-

temology is not conceptual analysis. Bayesian epistemology creates 

                                                 
34

 I don’t want to leave the impression that Riemann was simply playing about 

until one day, hey presto!, the general theory of relativity popped into view. 

Riemann’s geometry was a formidable piece of mathematics. It generalized 

Gauss’s work on surfaces to n dimensional manifolds. When n = 3, Riemann’s 

geometry is the result. Riemann didn’t think the 3D case described physical 

space or—since scaling up is a dubious assumption in physics—that it ever 

would prove to do so. Still, he did think it a physically possible hypothesis, 

hence a description of how the world might conceivably be. 

 



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323 

its own concepts, and makes true its own statements about them, 

graced by the entitlements of legislative postulation. Who knows 

what the future may bring? Who is to say that the concepts and 

theorems of Bayesian epistemology won’t find useful roles to play 

in the evolution of our understanding of human cognition? Who 

can deny that in thinking of knowledge and belief in Bayesian 

ways, further good ideas might also be thought up?  

 Whatever the future of synthetic Ns in philosophy, rosy or 
bleak as the case may be, there is a problem that they have yet to 

subdue. It is the problem with which we began. It is the problem of 

the gap between norm and practice. In its original form, it is the 

problem occasioned by the fact that if the theorems of the theory 

are true practice on the ground is bad or subpar or incompetent. 

Thus the pressure point of the matter was the truth or otherwise of 

its normative theorems. In the present situation, that is not the 

question. Conceived as synthetic theories, the norms of a N are al-
ways valid. They are true in virtue of the ways in which N has con-
structed its concepts. It may well be that behaviour on the ground is 

N-bad or N-subpar or N-incompetent. But this leaves it wholly 
undetermined whether it is bad or subpar or incompetent. The nor-

mative authentication problem remains unsolved. 

 The respects in which we may now accept the characteriza-

tion of Bayesian epistemology as mathematical—its use of mathe-

matical methods and its legislatively postulational character—

won’t make this problem go away. The same is true for N-theories 
in general. Behaviour on the ground may be N-bad. But what we 
want to know is whether it is bad. It would help enormously if 

philosophical N-theories had a paid-up account at the normative 
checkout counter. Unfortunately however, 

 

NORMATIVELY ROOTLESS: As presently conceived of, 

mathematical theories brim with new concepts and bristle 

with new truths. But they are theories with nowhere to go, 

and no future to achieve save for such further technical fi-

nesse that lies within the power of their creators and sanc-

tioned by the freedom to be as clever as they like.  



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324 

 

Let’s end this section where we began, with the last and utterly ex-

traordinary word going to Raiffa: 

 
If most people behaved in a manner roughly consistent with 

Savage’s theory then the theory would gain stature as a de-

scriptive theory but would lose a good deal of its normative 

importance. We do not have to teach people what comes 

naturally (Raiffa, 1961).
35

 

 

 

13.  Conclusion 

 

I close with a forecast and an observation. The forecast is that seri-

ous forces are now gathering with hegemonic intent, with a view to 

subjecting IL’s historical subject matters to the procedural rigours 

by which formal philosophy has achieved its current and future 

purchase in our intellectual affairs. The observation is that this is 

no more the right thing to do for theories of inference and argument 

than it has so far proved to be for theories of knowledge. What’s 

sauce for the goose is sauce for the gander. As we have it so far, a 

normatively presumptive mathematicization is the wrong sauce for 

epistemology. Enough said. Aux barricades! 

 

 

 

14.  A personal note  

  

I have had the pleasure of her acquaintanceship since I first met 

Trudy Govier in 1965 at the Summer Institute of Philosophy in 

Calgary, and have relished her scholarship in all the years that have 

followed. Informal logicians are more favourably inclined to appli-

cational rather than foundational matters, with Trudy leaning the 

first way, and I the second. Among her many achievements is a 

body of work on various aspects of social policy, unbettered by an-

                                                 
35

 The reference is to Savage (1954). 



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325 

ything of the same ilk done in the IL community, and ranking fa-

vourably with similar work from the other disciplines.
36

 This is ap-

plied informal logic with avengeance. Applications of such clout 

take much of the oxygen out of foundational enthusiasms. Such, I 

think, would be Trudy’s own view of the matter. Perhaps she 

would be right. I can’t help it. I salute her achievements in my own 

way. 

 

 

Acknowledgements: For astute criticisms and helpful conversation 

about mathematization, I warmly thank Artur d’Avila Garcez, 

Roberta Ballarin, Howard Barringer, Johan van Benthem, Paul 

Bartha, John Burgess, Roger Clark, Dov Gabbay, Stephan Hart-

mann, Jonathan Jenkins Ichikawa, Carrie Ichikawa Jenkins, Lo-

renzo Magnani, Chris Mole, Alirio Rosales, Steve Savitt, Ori Sim-

chen and Harvey Siegel. I am especially thankful to Rosales for 

several hours of vigorous conversion, and to Siegel for pressing me 

to expose something of my own views on the normative authentica-

tion problem. Thanks too, to editors Ralph Johnson and Tony Blair 

for their invitation to participate in IL’s tribute to Trudy Govier. 

 

            

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