Friday, 7 June 2013

Psychologism about Mathematical Entities

Empiricism is the epistemological doctrine that all concepts and theories, and justification for them, must be derived from experience. Rationalism is the epistemological doctrine that, in addition to experience, there are "innate faculties" of cognition. (I'm a rationalist, btw.) The debate between Rationalists and Empiricists goes back to philosophy's infancy, and will no doubt continue forever. And Psychologism about mathematical objects is the metaphysical doctrine that mathematical objects are mental entities.

Everyone, empiricists and rationalists included, has a problem answering the following kind of question:
(Q) How does one "experience" a number, say 23? 
It's very difficult for anyone to answer this. This is why empiricists have either been sceptics about mathematics, or have generally aimed to reduce mathematical objects to concrete entities, or have tried to reduce mathematics to logic, and claim that this is known analytically.

The question (Q) might be answered by saying:
(A1) One has some specific experience of some concrete token, which has 23 distinguished parts, say vertical strokes.
(A2) One has direct or indirect Platonic grasping of the number 23 itself, an abstract entity. 
(A2) is rather like Godel's answer. More exactly, Godel thinks we grasp abstract mathematical concepts. I'm perfectly happy with this and it's the answer I give. But if one's answer is (A2), then one is already assuming a form of Platonism. The main objection to this is that Platonic "grasping" is mysterious.

Psychologism aims to give an answer akin to (A1). And this approach was analysed by Frege in his discussion of Mill and Kant in Die Grundlagen der Arithmetic (1884).

So, e.g., think of the experience a human being has when looking at a token t of this following type
| | | | | | | | | | | | | | | | | | | | | | | 
[Update: as a commenter below noted, a similar kind of example would be geometric, involving a geometric line which is the concatenatation of 23 copies of equal unit length. This would involve 24 points $p_1, \dots, p_{24}$, with $p_{n+1}$ strictly between $p_n$ and $p_{n+2}$, and 23 congruent segments: i.e.,
$p_1p_2 \equiv p_2p_3,  p_2p_3 \equiv p_3p_4, \dots, p_{22}p_{23} \equiv p_{23}p_{24}$. 
I'm guessing, but I believe the analysis of this would lead to similar conclusions as given below.]

Ordinary human veridical visual/perceptual experience of the concrete token t delivers the judgement,
(1) There are 23 vertical strokes in token t.
To do this, the mind performs a mental computation:
Counting Computation (for a given input token t)
First: identify the relevant concept C.
Second: identify a discrete linear order R for these vertical stroke token parts of t.
Third: following the order R, make a sequence of noises, themselves arranged isomorphically to a canonical order $R^{\ast}$, as one examines each stroke in $t$, correlating noises for each stroke.
Fourth: The output is the final noise. 
The canonical relation $R^{\ast}$ = the order < on $\{1, 2, 3, 4, \dots\}$.
C = the concept "x is a vertical stroke token part of token t".
R = the physical relation "vertical stroke token part y of t is immediately to the right of vertical token stroke part x of t".
Perform this computation. The mental representation of the canonical order $R^{\ast}$ that we (users of English and English orthography with Arabic numeral-types) have learnt goes:
"1", "2", …. 
So, the output is some mental token representation of "23".

This computation can thought of as "copying" or "encoding" the physical relation $R$ on the token $t$, relative to the concept C, into one's mental representations, assuming we already have $R^{\ast}$. Hence, for it to work, the mind must somehow mentally represent the canonical order $R^{\ast}$, the concept $C$, and the relevant physical relation $R$ (on the given token). In fact, the essence of the counting computation is to establish that $R$ is isomorphic to an initial segment of the canonical order $R^{\ast}$, which is < on $\mathbb{N}$.

Abstractly put, the conclusion is:
(2) The physical relation R is isomorphic to the canonical relation $R^{\ast}$, restricted to $\{1,\dots. 23\}$. 
From the initial judgement (1), we get:
 (3) 23 = the number of vertical stroke tokens in the token t.
If, one the other hand, I do the corresponding counting computation with the experience generated by a token t' of,
 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 
then one gets the judgment (using a different concept, namely "x is a question mark token in t", and a different physical ordering relation $R^{\prime}$)
(4) 23 = the number of question mark tokens in the token t'. 
But, in (3) and (4), the numbers referred to are the same number: 23. Consequently,
(5) the number of vertical stroke tokens in the token t = the number of question mark tokens in the token t'. 
In a sense, what is crucial is simply:
(6) There is a bijection between {x | x is a vertical stroke token in token t} and {x | x is a question mark token in token t'}. 
So, as Frege explained, the inference leads to the assignment of the number 23 as the cardinality of the concepts (expressed by):
x is vertical stroke token in token t.
x is question mark token in token t'.
Is this what is meant by "experiencing" the number 23? If so, then it presupposes a lot of abstract machinery for the mind to already possess.

None of this establishes that the number 23 is a psychological entity. At best, one obtains the conclusion that experience of 23 involves a mental token representations of the number 23 (as well as mental representations of certain organizing concepts and relations, such as $C$ and $R$ and $R^{\ast}$). It does not imply that 23 is this, or some other, mental representation token. In fact, this would lead to contradictions, since, for smallish numbers, there are many such tokens, but only one type. And for largeish numbers, because there are no tokens of $n$, for $n$ sufficiently large.

Furthermore, as Frege explains, one does not "experience" the number 0; and one does not (probably, cannot, for biological reasons) "experience" large numbers, such as $10^{10}$.

For the case of 0, one can judge, e.g.,
(7) There are no horizontal strokes in token t
and thereby infer,
(8) 0 = the number of horizontal strokes in token t
But one can judge, by logic alone,
(9) There are no things not identical to themself, 
and so infer,
(10) 0 = the number of things not identical to themself. 
The reasoning (or mental computation if you like) here has a purely logically justified input, because (9) is a logical truth. No specific perceptual experience is required. (Caveat: unless one goes Quinian, i.e., epistemically holistic.)

The conclusion that this justifies is that:
Our knowledge of numbers is not obtained by (direct) experience. 
Rather, it is obtained by inferences, using more basic principles, assumed as primitive.
In particular, Frege pointed out, consider the principles:
(Comp-1)  There is an $X$ such that for all $y$, $Xy$ iff $\phi(y)$.
(Comp-2)  There is an $R$ such that for all $x,y$, $Rxy$ iff $\phi(x,y)$.
(HP) $\hspace{8mm}$ $|X| = |Y|$ iff $X \sim Y$. 
One can show that these assumptions lead to a theory, Frege Arithmetic ($\mathsf{FA}$) equivalent to second-order Peano arithmetic. (This is called Frege's Theorem.)

As Frege explained, one sees that one only needs to start with 0 (which does not require perceptual experience) and one can define the numbers. Roughly,
0 = the cardinal number of things not identical to themself = $ | \varnothing |$.
1 = the cardinal number of the concept being equal to 0 = $| \{0\} |$.
2 = the cardinal number of the concept being equal to either 0 or 1 = $| \{0,1\} |$.
and so on. 
Our experience might be anything one likes, and one could still infer the existence of, and standard properties of, numbers using (Comp-1), (Comp-2) and (HP).

There are further technical reasons why experience cannot make a difference. E.g.,
Theorem: (Conservation theorem for Frege arithmetic)
Let $E$ be a consistent experience statement. Then $E^{\ast}$ is consistent with Frege arithmetic, $\mathsf{FA}$. 
[Here $E^{\ast}$ is the relativization of quantifiers in $E$ to "$x$ is not a number".)

The proof is this: let $\mathcal{A} \models E$. Then $\mathcal{A}$ can be expanded to a model $\mathcal{A}^{+} \models \mathsf{FA} + E^{\ast}$.

So far as I can see, all of this makes Psychologism about mathematical objects a very dubious view, and explains its unpopularity amongst professionals working in the field.


  1. Another major view is that numbers represent physical quantities or lengths. The Pythagoreans would have argued that the number 23 represents "a length that is 23 units long." This view, although pretty intuitive for their work on geometry, hurt their ability to understand limits, so it is just as flawed.

  2. Thanks - yes, you're right.
    I should update a bit.



  3. Hi Jeff,

    "Is this what is meant by "experiencing" the number 23? If so, then it presupposes a lot of abstract machinery for the mind to already possess."

    Yes, but this doesn't preclude the question of how this conceptual machinery got into the mind in the first place, and this is largely an empirical question. Is the machinery in some sense innate, or did it have to be acquired by some specific cognitive processes? There's a vast literature on the onset on numerical cognition, and while there is still no agreement on some of the details, there's compelling evidence suggesting that learning to count is a socially learned skill, which in first instance goes very much along the likes of associating words for numbers to quantities of objects.

  4. Hi Catarina,

    I think cognitive psychology has moved on considerably since the 50s, and combines nature and nurture. So, yes, I think there are features of cognition which are innate, which operate on incoming information, and processes it.
    Do you mean cognitive processes to be mechanisms of association with sense data obtained via from social learning? The problem is that associationism can't explain much: memory, visual processing, linguistic processing, reasoning more generally ...

    "there's compelling evidence suggesting that learning to count is a socially learned skill, which in first instance goes very much along the likes of associating words for numbers to quantities of objects."

    Part of the problem with associationism (social constructivist variety) is with the notion of "socially learned": e.g., snails, chickens, etc., don't, probably can't, learn to count; primates probably not either. What explains this?
    The other part is with "quantities of objects", as Frege pointed out. For example, one counts the question mark tokens in a token of


    by mapping the things that are question marks (one needs to be able to recognize objects falling under this concept), relative to their spatial proximity relation, isomorphically to the canonical order


    an initial segment of the natural numbers, numerals for which we've learnt. And then output is the final term. So it's not really a quantity here. Do you mean a set? Rather there's a concept ("question mark token") and two orders, which are shown to be isomorphic by the mental computation.
    Then people "see" that the order on the set counted makes no difference to the output, and this allows them to see that what matters is the pairing-off, and not the order it's done in.

    But we shall probably never agree :)



  5. Hi Catarina,

    I've read various bits of cog psych research literature on numerical cognition over the years, but not since maybe 2004/5. On counting -- i.e., putting things in one-to-one correspondence -- the best work I've seen is by Brian Butterworth, e.g., "The development of arithmetical abilities" (2005),

    In the abstract:
    "The development of arithmetical abilities can be described in terms of the idea of numerosity – the number of objects in a set. Early arithmetic is usually thought of as the effects on numerosity of operations on sets such as set union. The child’s concept of numerosity appears to be innate, as infants, even in the first week of life, seem to discriminate visual arrays on the basis of numerosity. ...
    ... The evidence broadly supports the idea of an innate specific capacity for acquiring arithmetical skills, but the effects of the content of learning, and the timing of learning in the course of development, requires further investigation."



    1. There is still much dispute among developmental psychologists on these issues. My own preferred account of numerical cognition is by S. Dehaene: we have some sort of innate capacity for counting very small amounts (up to three or so), and an innate capacity for estimation of larger amounts. With training in the practice of counting, these two capacities 'merge' and give rise to basic exact arithmetic going beyond 'three'. I find this account compelling precisely in that it has a role both for 'nature' and 'nurture'.

    2. Hi Catarina,

      Yes, I think we discussed Dehaene a bit last year, or maybe longer ago!
      I think Dehaene is not discussing counting: i.e., putting sets in one-to-one correspondence, by enumerating them relative to some order. But Butterworth is clearly discussing counting in his work and in the article I mention above.

      In fact, I think Dehaene is discussing something like pattern recognition in physical/sensory aggregates. This may or may not have some genetic basis (almost certainly does, I guess). One would expect it not to be species-specific either. But this isn't counting, as far as I can see. It's more closely connected to the use of reals as measures of length (volume, etc), and not the natural numbers as cardinals.