"It is my contention that two quite distinct kinds of concerns have separately motivated accounts of the nature of mathematical truth: (1) the concern for having a homogeneous semantical theory in which semantics for the propositions of mathematics parallel the semantics for the rest of language, and (2) the concern that the account of mathematical truth mesh with a reasonable epistemology. It will be my general thesis that almost all accounts of the concept of mathematical truth can be identified with serving one or another of these masters at the expense of the other" (Paul Benacerraf, 2004: 403)

Benacerraf's above statements are intended to illustrate a problem with all contemporary theories of mathematical truth in that all theories must fulfil both concerns. (Benacerraf 2004:403-4).

In proceeding I will first investigate what is meant by both concerns, and then evaluate them as applied to Platonism and Constructivism where the strain explored by Benacerraf occurs. I find that Benacerraf is correct in that neither theory can live up to both of his concerns, but whether both concerns are wholly valid will be under discussion

It is the necessity of concern (1) that I find disputable. I acknowledge that there is a metaphysical problem that needs to be addressed in any valid account of mathematical truth but not that it needs to be dealt with in terms of a homogeneous theory of language. Referring to Wittgenstein I suspect (1) to be philosophically naïve. Rather than attempting to force mathematical language into the same structure as elsewhere in language, we should instead draw from the transcendental philosophy of Immanuel Kant.

Benacerraf's first concern is metaphysical: (1) outlines a desire that numerals behave in a semantically similar way to other nouns in language. Benacerraf's sentence structure examples are insightful: According the first concern 'There are at least three large cities older than New York' and 'There are at least three perfect numbers greater than seventeen' (Benacerraf 2004:405) should both share the same semantic structure

Benacerraf account of the first sentence's semantic structure is 'There are at least three Fgs that bear R to e'. (Benacerraf 2004: 405). That this interpretation is on the right track in general terms suffices for our purposes, and Benacerraf is right to dismiss further complications regarding the sentence's interpretation.

If the same semantic structure is accurate of the mathematical sentence then the term 'seventeen' should be understood similarly to the term 'New York', that being through reference. This implies there exists something akin to 'object reference' for numerals, which would necessitate some form of Platonism, which Benacerraf indicates agreement to by stating that when shying away from Platonism he shies away from supposing that the mathematical sentence shares the same semantic structure as the sentence regarding cities. (Benacerraf 2004: 406).

Benacerraf's second concern is epistemological: (2) refers to the need that the theory must preserve "the possibility of having mathematical knowledge" (Benacerraf 2004:409). This necessity can be seen in our knowledge of the statement 2+3=5. We do not merely have stronger belief in this statement than others, but find it impossible to doubt. If we know anything mathematical statements like 2+3=5 are amongst what we know, and thus our possession of such knowledge must be accounted for in any valid theory of mathematical truth.

I will proceed by outlining the problem by assuming both of Benacerraf's concerns are valid and necessary.

As previously noted, the first concern leads into theory broadly describable as Platonist; treating numbers in a way not dissimilar to objects. In treating numbers akin to normal objects we establish a 'causal relation' between our belief that 5+5=10 being true and the world itself in similar way to that of Hermione's belief that she holds a truffle and the location of the truffle in her hand within the world itself independent of her (Benacerraf 2004:421); for a belief to be true it must be in cooperation with some fact about the world itself. Platonism fulfils this metaphysical demand.

The problem with Platonic accounts are epistemological; we need to know how we come to any knowledge regarding these objects given that a Platonic account "places them beyond the reach of the better understood means of human cognition (e.g.., sense perception and the like)" (Benacerraf 2004:409). In the world I might see one banana, or three monkeys, but I have never seen, heard, felt, smelt or tasted 'one', or 'three', nor does it make any sense to talk of having done so. If numbers are separate from instances of numbers it is ambiguous how I know about numbers at all.

Plato's attempted solution highlights this issue: "The Soul ... having seen all things that exist, whether in this world or the world below, has knowledge of them all" (Benacerraf 2004:416). Plato recognised the epistemological problem and thus provided an account of knowledge of numbers via anamnesis. This sort of solution is obviously not suitable for modern appetites; Benacerraf only refers to it only in passing. Should such ideas be taken seriously we would require an account of how an immaterial 'soul' could perceive an immaterial object such as 'a number' despite modern explanations of perception and the retention and reminiscence of knowledge being rooted in biology.

Benacerraf describes our other contender as 'Combinatorial', wherein mathematical truth is discovered via its "derivability from specified sets of axioms" (Benacerraf 2004: 406) or from other syntactic truths about them. This provides a clear epistemology; in describing mathematical truth as derived from axioms or conventions of mathematics we provide an account of what we need to do in order to know a mathematical statement is true.

The problem is that we achieve the epistemological strength "at the cost of leaving it unintelligible how we can have any mathematical knowledge what so ever" (Benacerraf 2004: 403). Benacerraf's concern is that we cannot combine the syntactic approach with the Standard/Platonist understanding of mathematics.

Benacerraf requires that the truth of a mathematical statement amount to more than "their theoremhood in some formal system" but that we also show "the connection between truth and theoremhood" (Benacerraf 2004: 408). In short we must relate correctness within our theoremhood to the world.

However, if we hold mathematical statements to be true purely because of their derivability in a theorem then we cease to be Platonists as their truth is not defined by reference to any such platonic objects. If we lack reference then all mathematical truth is discovered analytically rather than synthetically, which would mean that mathematical propositions and the concepts they use are merely systems within our minds, which would make us Constructivists. Benacerraf refers to this as "the realisation that mathematics is a child of our own begetting" (Benacerraf 2004: 417)

If we define mathematical truth as above we lose mathematics' causal relationship with the world; Hermione's belief that she holds a truffle is related to her actually holding a truffle, our belief in a mathematical statement however is related to its correctness within a theoremhood not to an independent fact about the world itself. However, when I say that 5+5=10 is true, I do not mean that it is true within a system, I mean that it is true of the world; any valid theory of mathematical truth must account for this.

When explaining mathematical truths through reference the above demand for a causal link is provided for by the clear relationship between a mathematical proposition and the independently existing platonic numbers it refers to. If we lack this concept of reference we set mathematics adrift; mathematical discovery thus becomes "seldom a discovery about an independent reality" (Benacerraf 2004: 417). Hence if we follow through with this sort of Constructivism we are led away from mathematics as being based in the external world; instead it becomes based in our own minds, which would most likely implies an intuitionist basis for mathematical truth wherein mathematical processes become entirely internal and there seems no obviously knowable relation to the world at all. However if we fail to relate these 'truths' to the world, which they are supposedly true of, we lose the right to call them true.

I will now begin a critical evaluation of the two concerns' legitimacy and the objection they support. I hold it true that we need both an account of how mathematical truths are knowable, and an account of how these truths relate to the world, and are thus in some sense 'true of the world'. As I will defend a Constructivist account of mathematical truth, I thus shall need to supply for a metaphysical underpinning for Constructivism.

The fault with Benacerraf's depiction of the problem is the demand that this metaphysical underpinning be achieved via applying the same semantics found elsewhere in language in mathematical language. Hence I hold that Benacerraf's requirement that "the propositions of mathematics parallel the semantics for the rest of language" (Benacerraf 2004: 403) is misguided. I thus oppose the injecting of the semantics we use elsewhere in language into mathematical language.

The above desire is opposed at length in the later philosophy of Wittgenstein; instead of a homogeneous theory of language he describes language via the following metaphor: "Our language can be seen as an ancient city: a maze of little streets and square, of old and new houses, and of houses with additions from various periods…" and so on. (Wittgenstein 1972: §18) Such a language could hypothetically be mapped but it could not fit under one general theory (no more than the different areas of London could fit under one general organisational theory)

Wittgenstein has valid reasons for believing that language systems of numbers and normal nouns work differently, which he illustrates through the concept of 'language games' such as his building language (Wittgenstein 1972: §8). In such thought experiments it is noted how different parts of language are built for different functions and usages, thus it should not be surprising that they work differently. As numerals are very different in function to nouns like 'hammer' or 'spade', it is not surprising they cannot be explained through one homogeneous theory.

If Wittgenstein is right then the notion of treating the role of a numeral in a sentence such as 'seven is larger than three' as being at all akin to 'London is larger than Southampton' is entirely misled. In comparing mathematical propositions to propositions about physical things in the world we would be fundamentally confused.

However, whilst Wittgenstein may be right in that different parts of language work in very different ways, and hence lack shared semantic structures, we do need that aforementioned metaphysical weight for Constructivism to be a valid theory. However, I hold that complete solace can be found in the transcendental philosophy of Immanuel Kant.

I will not indulge in the particulars of the Kantian approach to mathematics; it would take too long and is unnecessary for my purposes; I only require the distinction between the phenomenal world constructed by our minds and that of the noumenal world 'in itself' of which the phenomenal world is a representation (Kant 2002: 143-152), as well as the concept of 'Modes of Representation' (Kant 2002:31-32)

It is not a radical but instead a tautological thought that our perception of reality is phenomenal; phenomenal only denoting 'perceived reality'. That our minds are born with the concepts of space and time and that we do not reason or observe our way to knowledge of them should equally not be radical (Kant 2002:32-33)

Kantian metaphysics retains reference for normal items; 'New York' has a reference as an object within the phenomenological world. When stating truths regarding cities we reference actual objects.

Other statements do not reference objects in the phenomenological world but instead describe general truths about phenomenological reality. Such truths describe 'Modes of Representation' rather than 'items of representation'.

Space and time are modes of representation and not items of representation but they are real: "Time is certainly something real, that is to say, it is the real form of the internal Intuition" (Kant 2004: 39). For Kant geometrical and arithmetic truths are true about the modes of representation and thus do not refer to any item of representation but describe the structure items are represented within, enough for us is that we can plausibly discuss mathematics as being true of the structure of reality rather than objects within it, and therefore true of the world as well.

Such truths are about "subjective reality in regard of internal experience" (Kant 2004: 39) and thus are constructivist in some sense, but our subjective reality is still related to reality in itself (noumenal reality) due to our subjective realities being representations of noumenal reality; thus mathematics is not left metaphysically adrift. Whilst numbers are neither noumenal or phenomenal entities, they describe the system of representation by which independent reality is represented.

Similar truths are found in the ordering of colour shades by lightness; similarly the names of colours refer to no particular thing and play a distinct role in the sentence that is unlike other nouns.

Transcendental philosophy supplies us with ultimate assurance of the truth of mathematics regarding reality; although in exploring mathematical truths we do not discover facts about anything in the world, we do still discover facts about the world.

Benacerraf is correct regarding the epistemological problem with Platonism, and the inability of syntactic theories of mathematical truth to be combined with standard/platonic conceptions. He is incorrect that they are required to be compatible with Platonism be in order to be related to the world. As noted previously, a statement can be true of the world without having any particular reference to anything in the world. Hence with Kantian metaphysics we are offered an alternative to the Platonic understanding of how mathematics relates to the world and thus we can avoid the problems Benacerraf raises.

Benacerraf, P (2004) Mathematical Truth, as in Philosophy of Mathematics, Selected Readings, edited by Benacerraf, P and Putnam, H, Oxford: Oxford University Press

Wittgenstein, W (1972) Philosophical Investigations, trans. G.E.M Anscombe, Oxford: Blackwell & Mott, Ltd

Kant, I. (2002) The Critique of Pure Reason, trans. F. Haywood, London: Living Time Press

Despite the above consisting of merely the work of a graduate, nor a post-graduate nor any professional academic, I felt it was important to provide proper referencing information for anyone who might possibly require it.

I do this partly because I used to find it annoying to read pages online and struggle to find the information I required in order to reference the article, and partly because plagiarism is a very serious issue in many educational establishments.

Hence for the sake of the possibility of someone needing it, here would be a typical format for referencing this article. You should check with your university department in order to get precise instructions of how to format your bibliography.

Hankins, J. (2007) Numbers as Modes of Representation, not entities, Available from the World Wide Web: http://www.skeletalroses.co.uk/html/articles/metaphysics_spatial_multiplicity.htm