CONTENTS

1   INTRODUCTION

2   SCIENCE OF SENSE

3   APPLIED MATHEMATICS

4   AXIOMS AS DEFINITIONS

5   ALTERNATIVE THEORIES

6   CONCLUSIONS

5   ALTERNATIVE THEORIES

The SS thesis is a high-level characterisation of mathematics, and it is neutral with respect to many of the great questions which have dominated the philosophy of mathematics during the twentieth century. Instead what it does is transform them into questions about the realm of sense, its ontological status and out epistemological relation to it. This is illustrated by the following examples.

OBJECTIVITY

Are mathematical entities objectively pre-existing, or are they a free creation of the human mind?

becomes

Are the senses of linguistic entities objectively pre-existing, or are they a free creation of the human mind?

CONSTRUCTIVISM

Which logic is appropriate to the business of mathematics: classical or intuitionistic?

becomes

Which logic is appropriate to the investigation of the realm of sense: classical or intuitionistic?

LIMITATIVE THEOREMS

What do the limitative theorems tell us about mathematical truth and mathematical knowledge?

becomes

What do the limitative theorems tell us about the utility of the formal method for exploring the realm of sense?

To repeat, the SS thesis does not immediately settle these questions. The fruitfulness of the thesis will depend to a large extent on the utility of this way of reframing the questions.

By contrast, the SS thesis is in direct conflict with traditional accounts of mathematics to the extent that these accounts make reference claims, that is, where mathematics is supposed to be about some sort of mathematical entities, the way physics is about physical objects. If we follow Frege in saying that indirect references is about sense, then we have to reformulate this distinction by saying that traditional, referential accounts of mathematics claim that mathematics is in the business of direct reference, whereas the SS thesis says that mathematics is concerned with indirect reference. In what follows we challenge the SS thesis by examining the direct reference claims traditionally associated with mathematics.

5.1 Analytic Reference

The SS reformulation of the question of objectivity leads us towards the question of the objectivity of sense. The argument for objectivity was sketched very briefly in the Introduction, and will be examined in more detail in a separate essay. However we also have to consider the aspirations of some parts of mathematics (or of some mathematicians) to deal with a Platonism of reference. Much of pure mathematics contains the names which purport to refer to objects called numbers. No account of mathematics can ignore these reference claims. Moreover the reference seems to be to objects other than the contingent physical objects of which we appear to obtain knowledge through our sensory apparatus (more generally, through our participation in the physical world). There is a strong intuition in mathematics that not only are the numbers different from the physical objects, but they are different from all contingent objects. That is to say, the existence of numbers is a logical necessity. Mathematics as the science of numbers is then a Science of Necessary Reference (SNR).

Splitting up the intuitions into two steps like this shows that Platonism is wider than SNR. There can be a Platonism of contingent objects. These objects are Platonic in the sense that they are not physical: they do not exist in space and time, and are not to be discovered by us via sensory apparatus. The distinction can be seen in attitudes to set theory. Fregean logicism claims to be an SNR; sets and all the objects constructed as sets, including the different forms of numbers are logical notions, and their references is guaranteed in the same way that the reference of the sentential connectives and the quantifiers are guaranteed.

Axiomatic set theory by contrast is a Science of Alternative Reference (SAR) - that is alternative to the reference to things in the physical world. It describes contingent Platonic objects. One can then imagine universes completely without sets and universes containing nothing but sets. There can be universes in which the Axiom of Choice is true and those in which it is false. At this point one is left wondering what could be the epistemological basis for determining which universe we happen to live in. Axiomatic set theory, as suggested by the appearance of the word "axiomatic" in its name, appears to collapse back into being part of the Science of Sense, namely the study of the senses of different set-notions. Even here we are studying the senses of certain features of sets - the sense of "x is a set" is not fully determined because we are left unclear as to how we ought to recognise a set were we to encounter one, or even in what context we might do so and what such an encounter would consist in.

We return to the possibility of a contingent but non-physical reference in the following sub-section. Here instead we continue the examination of the possibility of logically necessary reference. Let us start by looking at numbers; the extensive (to put at mildly) reference to numbers throughout mathematics is the main motivation for believing in necessary reference.

Traditional mathematics started out with two broad areas of study: number and space. When Descartes arithmetised geometry, this was reduced to the study of number. Abstract mathematics, by contrast, is supposed to study in a systematic way whatever senses are deemed worthy of such study. In practice however, most of the structures studied by abstract mathematics are those inspired by the study of numbers. Some, such as metrics and measures, show this by explicit reference to real numbers. Even in topology, which makes no such reference in its axioms, the aim is to generalise ideas first encountered in real analysis. Topology is truly abstract in that specifically numerical features have been abstracted away and one is left with a structure which can be studied without any reference to number. But once examples are given, it becomes clear that number is what the developers of topology had in mind. In a similar way, abstract algebra is inspired by the structures of the arithmetics of the various number systems. Abstract mathematics has not achieved its full independence from its traditional, Platonic parent.

So, what are we to make of this apparent reference in mathematics to these entities called numbers? From an SS perspective we could say that pure mathematics studies only the senses of the number-names, without commitment to their reference. This will however not work for the many times we make use of these names when talking about the world, from the simple use of cardinal numbers to count objects to the use of complex-number-valued wavefunctions in quantum mechanics. If we are to take these usages seriously as part of our referential discourse we must either declare a commitment to the existence of these objects, or reconstruct our language in a manner which avoids referring to them.

Of course, for almost all practicing mathematicians it is clear what sort of objects these numbers are. They are, like everything else in mathematics, special cases of sets. If we approached modern pure mathematics uncritically, like anthropologists studying the beliefs and rituals of some newly discovered culture, we would say that mathematics is set theory. Set theory can fulfil this role because at its heart it possesses the universal object forming operator. To be sure, the operator had to be made less universal than originally conceived, so as to avoid paradoxes, but it can still generate all the objects a mathematician might desire.

It is wonderful to live and work in Cantor's paradise, but if we take ontological commitment seriously, then there remains the nagging doubt: why should we believe in the existence of all these objects, all these sets? If some mathematical atheist came along and denied their existence, what contradiction would he or she incur? We seem to have to fall back on Quine's assertion that sets are simply to useful not to be believed in. This rests however on the dubious argument that whatever is useful must exist, an argument known to fail in the physical world. (In the desert, the obvious supreme utility of water does not suffice to provide water.)

This theme is developed further in the essay "Against Sets" (not yet available). Here we continue with the theme of the reference of number-names. It would be possible to believe specifically in the existence of numbers without being committed to the ontological extravagance of set theory. But what is it that pushes us towards belief in numbers. The first thing we ever use numbers for is counting. And Frege has shown us that the basic concepts of cardinality are captured by second-order predicates, the numerical quantifiers, and these can be constructed using the resources of quantifier logic. Let us write "there are exactly n Fs" in Fregean form as:

nx F(x)

so that:

and so on inductively.

This seems to solve the problem exactly. Cardinality is not about these strange objects called numbers, but instead is a property of concepts. Moreover we do not have a problem with ontology. As long as the logical constants: the sentential operators, the quantifiers and the identity relation have reference, then so do all these numerical quantifiers. Being part of logic, these constants are reasonable candidates for having logically necessary reference, and so this logical necessity is also inherited by the numerical quantifiers. (Note that this "natural" account of cardinality does not cover our other main use of numbers, the use of real-valued quantities in physics. )

Our joy and relief, as Quine points out, are short-lived, because we rapidly realise that all the things we are used to doing with arithmetic are not available to us. We do not have the resources to express sameness of number, or to combine numbers additively. If we are to remain within the bounds of first order predicate logic we are driven back to treating numbers as objects. The alternative is to introduce the possibility of identity and quantification at all levels in the Fregean hierarchy. As well as the identity of objects:

x = h

we have identity of first-order concepts:

and second-order concepts:

and so on. (In the third identity, P and Q are argument-places for second-order predicates, and f is a first-order predicate dummy variable.) The third identity is the one we need to express sameness of cardinality. Likewise we have hierarchies of universal and existential quantifiers, the nth quantifier being a predicate of order n+2.

This proliferation of entities of higher orders raises a number of questions.

• Ontology - What grounds do we have for thinking that the higher order quantifiers and identity relations have a reference? However this question can be turned around. Why should we believe that the first order expressions which are the basis of predicate logic have necessary reference and not accord the same status to the analogous expressions at an arbitrarily high order? It is not on the grounds of utility that we want these things; they are forced on us by our ontological commitments. If we have reference at level n, then surely we must have the possibility of sameness of reference, expressed by the level n+1 identity relation. Similarly the notion of existence associated with reference at this level forces us to have an existential quantifier at level n+2. (Note however that our commitment to the reference of all these entities does not require us to use them. We can stop ascending the hierarchy whenever the expressions cease to be useful.)
  
  
• Expressive power - As we go up the hierarchy, the mechanism of argument places and dummy variable becomes increasingly cumbersome. If the cardinal numbers are really these second level concepts, it would be seriously tiresome to do arithmetic with the third level identity, operators mapping two second level concepts onto one, and quantifiers which were fourth level predicates. School children would not be able to start counting and doing simple arithmetic until they had first mastered advanced logic. What we appear to do in practice is to build a zeroth level model of arithmetic, in which the numbers are objects. When we use the numbers to count, we use the numerical quantifiers (though for most people this is done without understanding), but when we want to do arithmetic, that is, operate entirely within the same level, then we work with the universe of numbers as a model, counterfactual but with a known link back to the reality of the numerical quantifiers.
  
  
• Algorithmic power - It is all very well to introduce this additional expressive power with the new, higher-level expressions, but to do serious mathematics with them we also need the associated rules of logical transformation. It is a well-known result that the power of formal logical systems decreases with increasing expressive power. If however this is genuinely what arithmetic is about, the this may be a burden we simply have to learn to bear. We would simply have do the best with what algorithmic power is left to us.

Having glanced at the aspirations of the numbers to be entities (at whatever level) with a necessary reference, let us move on to general question of when there can be necessary reference, and the extent to which the study of expressions which refer in this way can be identified with mathematics (or with a subset thereof). We recall from the discussion of partial orderings that when we shifted our attention from the ordering relations themselves to the second level property which says of a relation that it is a partial ordering, we then found that all the remaining constituents of the expression were constants from quantifier logic. Relative to the necessity of the reference of these constants, such expressions do not only define a sense which we want to study, but also a sense which must lead to a referent.

Can we turn this relative necessity into an absolute one? Let us start with the basic notion that the primary purpose of reference is to talk about things which are contingent. However, once we have built a structure of saying to do this, there emerge some truths which are about this mechanism, and not about the underlying world being talked about. The primary reference is of names, referring to the objects in the world. At the other end of the reference spectrum are the truth-values, whose existence is purely a consequence of what we have to do to say something about anything. From these follows the reference of the truth-functional sentential operators. Identity and existence are also about reference, so their reference is also a necessary consequence of the nature of the machinery. Frege showed us that once we have all of this, then there is a huge range of interesting truths which are about entities whose reference is necessary.

Out of these ideas come two conjectures.

Conjecture 1 - there are no necessary objects. Stated more accurately this says that there are no names with necessary reference. The idea behind this conjecture is that names are at the contingent end of the spectrum of reference; they have nothing to do with the mechanism of saying. Only incomplete expressions are candidates for having necessary reference. If it were shown to be true, this conjecture would have the dramatic consequence that the claims of set theory to speak of logically necessary objects, namely the sets, would be false.

Conjecture 2 - the subject matter of mathematics is necessary reference. Necessary reference is a property of sense, so its study is part of mathematics viewed as the Science of Sense. The examples studied above, in which all expressions were removed until only the logical constants remain, hint at the possibility that this part of mathematics is in fact the whole.

If both of these conjectures are true then mathematics is the study of certain incomplete referents. Usually these will be of second or higher level. When we looked at ordering relations, the subject matter was of type 1S2SNN, that is, a second level predicate applicable to first level relations. When we looked at cardinality, the expressions were 1S1SN second level predicates. The general problem with this sort of mathematics is that our study of anything is gravely hampered if we are not allowed to quantify over them.

Where does the account of mathematics emerging out of the above discussion (albeit not yet in sharp focus) leave the logicist contention that mathematics can be reduced to logic? The original logicism was reckoned to have been lost with the wreck of naive set theory. Naive set theory might reasonably have been thought a natural extension of logic, but ZFC seems too arbitrary to count as such. If mathematics needs set theory (or is identical to it), and set theory is not part of logic, then mathematics is not part of logic. Quine saw mathematics as adjacent to logic, the border being crossed when the notion of set (in whatever sanitised form one chooses) is introduced.

What happens to this argument if we replace set theory as the foundation of mathematics with the SS thesis, with the senses under study being expressed by higher level incomplete expressions, some or all of which may be entirely made out of logical constants? Does not this return mathematics to the status of being part of logic, or even identical to logic? The distinction between the two subjects can probably be expressed as follows: logic is about entailments, mathematics about axioms. This distinction is parallel to that between modal and historical sciences. Logic is concerned with the transformation rules, which are the "laws of motion" of thought. Mathematics is concerned with taking specific "initial conditions" - namely the axioms - and then seeing what the laws of motion do with them. All this bears out very strongly what Dummett said about axioms in formal logical systems. He argued that Frege's analogy between logic and axiomatic mathematical systems, such as Euclidean geometry, although fruitful to begin with, ultimately was misguided. A better picture of what logic is about is provided by systems based entirely on transformation rules.

The purpose of this essay is to explore ways of making better sense out of what mathematicians, pure and applied, actually do, and not to suggest that it be replaced by something completely different. And yet we return to the anthropological fact that most mathematicians, at least within the pure discipline, do set theory, whereas here something quite different is being suggested as what they should be doing. The resolution may be something like this. Rather than struggle with all the higher level expressions, mathematicians use set theory to "flatten" the whole structure, so that everything can be done with objects, and therefore with the lowest level quantifier logic Frege bequeathed to us.

5.2 Contingent Platonism

Now let us consider the possibility that mathematics is about an alternative realm of reference, neither the spatio-temporal world of the sense organs, nor the logically necessary realm of the referents of expressions formed out of logical constants. We participate in this world through our faculty of mathematical intuition; this is our epistemological link to the mathematical world, just as our sensory apparatus is our link to the physical world. It is part of the phenomenology of human existence, the argument goes, that we have a mathematical side to our being as well as a physical one.

Starting from a purely phenomenological perspective, with no a priori prejudices about what worlds there are out there, or what epistemological faculties human beings might have to connect them to these worlds, we have no immediate reason to reject mathematical intuition as a candidate for evidence of reference. There is, in particular, no reason to accept sensations of the physical, and at the same time reject intuitions of the mathematical, for example, on the grounds that the subject matter is not physical and therefore not real. (This works both ways. Those of a Platonic disposition might say that the subject matter of mathematical intuition is more real than the objects of sensation, because they are timeless, not subject to the mutability and decay which persistently infects physical objects. This is a vulgar prejudice - however exalted its origins - which has no place in ontology. There are no a priori grounds for supposing reality to be timeless.)

A caustic criticism of such an introduction of a new faculty to explain a putative new type of knowledge is provided by Nietzsche. In his wide-ranging polemic against "the Prejudices of Philosophers" at the beginning of "Beyond Good and Evil", he attacks Kant for claiming to have discovered a faculty by means of which human beings gain knowledge of the synthetic a priori. He compares this with Molière's satire of the doctor explaining the power of opium to induce sleep in terms of its "dormative virtue".

This criticism is valid if we were introducing the talk of mathematical intuition as a putative explanation of why there is a mathematical reality and of how it is we get to know about it. All that is being attempted here is, however, only a description of a piece of phenomenology. There appears to be this body of truth which is separate from truths about the physical world, and of which we gain knowledge not through the operation of our sensory apparatus, but simply by thinking about these matters. We are at this stage of description as unprejudiced as is humanly possible as to what the ultimate explanation of these phenomena might be; it is central to the discipline of phenomenology to use language as uncontaminated with explanation as possible. We remain open to the possibility that the ultimate explanation might do away with talk of an alternative ontology and an alternative epistemological faculty. The Science of Sense explanation of mathematics is precisely an attempt to do so.

There is a strong analogy here between the supposed mathematical intuitions people have and reports of religious experience. It is clear phenomenologically that people do have religious experience. The problem lies with the interpretation of what it is an experience of. In the case of religion, the usual interpretation in terms of supernatural beings is suspect because of the wide divergences between the reports given by different religions. With mathematics we seem to be on sounder ground. There is a much greater consensus between mathematicians as to what counts as the subject matter of mathematics and what are the truths about this subject matter, even taking into account the schism between the rococco Catholicism of ZFC and the austere Calvinism of constructivism.

Still looking at this supposed mathematical faculty at the phenomenological level, we must note a strong disanalogy with sensation and with religious experience. Almost all people have sensory experience, and the absence of it is counted a very serious defect in the physical grounding of the person. Religious experience is widespread across all cultures, though its exact nature is highly culturally dependent. By contrast, comparatively few people have much in the way of mathematical intuition. By this intuition I mean much more than mere numeracy (which in itself is in less than abundant supply), but rather the whole habit of thinking in terms of axioms, theorems, generalisations, proofs and counter-examples. This mathematical faculty lies at the outer margins of human ability, a piece of data which has to be taken into account in any explanation of the phenomenon.

As we begin to explore the phenomenology further, looking to see if we can crystallise out of it a credible ontology, there are two questions to be asked: objective or subjective, contingent or necessary? As a means of addressing the first question, let us compare with two standards: physical reality as the epitome of objectivity and economic value as the epitome of subjectivity. In the former case, what there is seems to be stubbornly independent of what we believe about it, while in the latter the reality is notoriously sensitive to what people collectively believe about value. Looked at in this light, if there is an alternative realm of reference underlying mathematics, it is clearly closer to stubborn objectivity of the laws of physics than to the volatility of stock market prices.

Is then the nature, and indeed the existence, of this alternative reality contingent or necessary? Again let us compare with standards, taking logic as the epitome of necessity and the laws of physics (or indeed the history of the physical world) as the epitome of contingency. The notion of a realm of alternative reference starts out with an inuitive bias towards the contingent. Just as there happens to be a realm of physical reality, so we discover this second realm, the realm of the mathematical. As a further contingency we discover links between the two realms, links which manifest themselves in the utility of applied mathematics, our use of mathematical discourse to understand better the nature of the physical world.

However, taking the phenomenology uncritically, mathematics feels necessary. We expect that in all possible worlds the same laws of numbers would apply, whereas the laws of physics could be quite different. This intuition can be turned into a question: if the laws of mathematics were contingent, upon what would they be contingent? Perhaps, like the laws of physics, they would be an irreducible contingency, contingent only on the nature of the world, in the circular sense that they define the nature of the world. This would make them part of physics.

Alternatively, seen as creations of the human mind, the theorems of mathematics might be contingent on the nature of the human mind. For this to be a true contingency, the nature of the mind would itself have to be contingent. There would have to be radically different possible forms of mind, each with its own associated mathematics. If all the properties of mind were constrained by the concept of mind, then the mind-dependent realm of mathematics would likewise be a necessary consequence of what mind must be. Conversely if the nature of mind is contingent, then this might be a timeless, modal contingency, varying only between possible worlds, or it might be such as to allow for the evolution of mind through time. In any case, the onus is upon those who advocate the contingency to show what an alternative possible mathematical world would look like, perhaps on the basis of an alternative possible form of mind.

Having examined the phenomenology of alternative reference, and having concluded tentatively that it looks like a phenomenon with an objective basis, we can begin to explore what might count as the split of the phenomenon into ontology and epistemology. This is where constructivism offers itself as a candidate. It produces a notion of construction and claims that this is the mode by which mathematical truth can be accessed.

Within constructivism there is a tension between "can be constructed" and "has been constructed". Which is the condition for proof? If we take the latter path, this makes mathematical truth historical. Then we have to answer the questions: can a construction cease to exist by virtue of having been forgotten? does it have to be in someone's mind, or does being on paper count as well? It is useful to make a comparison here with physical constructions. These depend upon human effort for their coming into being, and can be destroyed by human effort, but in between they exist objectively. While they exist they do not depend upon continuing human thought or belief (unlike, for example, the value of money). They are objective, and are constructed and destroyed only by human participation in the world of the objective.

The simplest resolution of this tension within intuitionism is as follows. The ontology is based upon "can be constructed". The results of mathematics exist timelessly within the modal space of possibilities. The epistemology is then based upon "has been constructed". We get to know about the truth of some mathematical statement either by performing the appropriate construction ourselves, or going through the construction someone else has carried out. If the construction is lost, then the statement is still mathematically true, but knowledge of this truth has been lost to humanity.

Although the SS thesis also sees mathematics as the study of a different realm of reference, it is quite different in its thrust to the notion of alternative reference. The realm of sense is an additional reference rather than an alternative. It is firmly built upon the primary realm of reference, whatever this may turn out to be. Its existence and contents are logically necessary, being derived from the mechanism of saying rather than depending upon the details of the contents of primary reference. Our intuitions of an alternative realm of mathematical reference can then be interpreted as the result of an imperfect perception of the realm of sense and an incorrect understanding of that perception.

In the final analysis, the alternative reference picture is too restrictive a view of mathematics. If mathematics were only a study of number systems, then it would be plausible, but modern mathematics is not so constrained. It has the right to study any structure it cares to dream up and consider worthy of study - even if in practice the number systems are such a rich source of structure that many mathematicians do not stray far from it. The contingency associated with the idea of alternative reference is the problem - it is a crucial part of mathematics that, given some particular mathematical system, mathematicians will ask, "what if we change this bit, what if we generalise this notion, what if we weaken or strengthen this assumption?" This instinct for a remorseless searching of alternative possibilities is what distinguishes mathematicians from mere human beings. This inheritance of the total field of logical possibility is captured by the SS thesis, better than some more restricted field of alternative reference. If constructivism can access large areas of this field, then it can be re-interpreted as being a candidate, not for the means of access to an alternative reference, but for the title of the correct method for carrying out the scientific study of the realm of sense.

5.3 Formalism

In our search the reference of mathematical discourse three main alternatives have been explored so far:

• logically necessary reference,
• alternative contingent reference,
• indirect reference (the SS thesis).

The fourth alternative is formalism, which can be included in this framework by saying it is the thesis that pure mathematics has no reference. It is instead the systematic study of various types of formal manipulation carried out upon strings of symbols. It is not that these strings have a sense which is then found to lack reference to the world; rather the strings do not even have a sense. In fact they are not discourse at all, although they bear certain formal similarities to formalised language, just as the transformations of strings bear formal resemblances to the truth-preserving transformations of logic. We do not encounter language until we rise to the level of meta-mathematics, where the reference is to the strings and their transformations.

When we turn to applied mathematics, this description has to be modified. The strings then are interpreted discourse on the periphery, but the connections between the interpreted parts of the periphery are not interpreted. In the formalist account of physics, certain terms refer to the experimental set up. The uninterpreted body of the mathematical theory connects these terms to certain output numbers elsewhere on the periphery which are interpreted as the (numerical) outcomes of the experiments. The theory has been falsified if the outcomes of the experiment differ from the corresponding predictions of the theory. In this strict formalist account, no attempt should be given to assign a reference to the intermediate parts of the theory, and in particular, the theory should not be seen as referring to an ontology lying behind the experiment, providing a causal link between set up and outcome.

Formalism within mathematics is motivated by the success of the formal method. Although Frege was strongly opposed to formalism, the success of his Begriffsschrift programme, of making the sense of sentences explicitly visible in their syntactic form, made formalism possible. Once this is done, logical transformations can be carried out purely sytactically without regard to the underlying sense - and this can be done by syntactic engines which, as Searle points out, have no contact whatever with the semantics. Because the theory of semantics, of linking language to some exterior world, is difficult, and because this is especially so with mathematics, the temptation is great to stay entirely within the realm of syntax.

As for applied mathematics, and in particular, mathematical physics, I conjecture that the historical motivation for a formalist approach is as follows. Ever since Newton introduced action-at-a-distance forces, and especially since these were added to by Faraday and Maxwell, the purity of the preferred ontology of physicists, namely atomistic materialism, has been under threat. However, generating a new ontology would have taken physicists out of the safe realm of physics into metaphysics, where, it seemed, nothing could ever be proved or disproved. Following Newton's hypotheses non fingo they followed, to a greater or lesser extent the practice of using the formalism of their theories without attempting an ontological interpretation (at least as the official part of physics, as opposed to what might be discussed over drinks in the pub). The gap between the success of formalism and the failure of ontology was maximised with the advent of quantum mechanics, and thereafter what I like to call ontophobia became part of the standard mental toolkit of the practicing physicist.

The science of formal systems is a legitimate enterprise (though some may question whether there is a point to it, apart from as an amusing game, like chess and cross-word puzzles). If done mathematically it might be seen as being a branch of applied mathematics, a possible physics of certain conjectured objects, not to mention the actual emergent physics of the syntactic engines. The question is whether it is an adequate characterisation of the whole of mathematics. When we do various forms of geometry we appear to be talking about spatial relations, even if only possible ones, not realised in any actual space. When we are doing arithmetic most of us believe we are talking (in some way, even if not using numbers as objects) about the cardinality property of concepts. To be sure we can do purely formal transformations on the numerals, without any reference to what they mean. But at each stage the meaning could be contacted, and the transformations derive their validation from the preservation of truth.

This raises the question of whether there can be formal manipulations on interpreted formulae which then take us into a semantically uninterpreted realm. A putative example comes from quantum mechanics. Starting with the position-space wavefunction, one can perform on it a Fourier transformation. In the conventional interpretation of quantum mechanics the result has a clear interpretation: it is the momentum space wavefunction. It is the mapping from possible momenta onto amplitudes for transitions to those momentum eigenstates. In my approach to quantum mechanics this interpretation is rejected. Instead I say that the Fourier transform is a formally useful but semantically meaningless piece of additional formalism. For me to make this move I have to envisage that formal transformations can take us from the interpreted to the uninterpreted part of the formalism.

Even if my point about the momentum wavefunction is incorrect, surely there are vast numbers of other formal manipulations one could perform, where one is in no way obliged to find interpretations for the results. Even if the Fourier transform is interpreted, there is the Laplace transform of the wavefunction which may be hard to interpret within quantum mechanics, even if performing the transformation makes it easier for us to solve important differential equations. Having solved these equations we can then re-interpret the end result, making useful predictions about the subject matter of quantum mechanics. Likewise even if I do not want to interpret the Fourier transform of the wavefunction, I still recognise it as a useful formal device for predicting the results of momentum measurements.

The contrary view is that if the semantics is organically attached to the initial parts of the formalism, then a new semantic interpretation will be automatically generated for the results of any formal manipulations of that initial core. The transformations will pull the semantics along with them. Of course this does not entail that the new meanings will always be useful, and we may be content with just the formal results, as a road to getting somewhere semantically useful in the end, without having to understand just what the intermediate steps mean. It will however always be possible to use the initial interpretations and the transformations to derive a meaning for the new entities. At very least this might simply invoke the inverse transformations and the original meanings. Our problematic Laplace transform of the position wavefunction would be interpreted as the function which, when subjected to the inverse Laplace transform, yields the position wavefunction.

In this example from theoretical physics it must be borne in mind that there are two layers of interpretation. Firstly the marks on the paper are interpreted in terms of real and complex numbers and associated functions. Then these numbers and functions are interpreted in terms of the probabilistic dynamics of particles and fields. Even if we run out of physical interpretation the formalism still means something mathematical. Suppose I want to derive some property of a wavefunction, and find that the derivation is easier (or possible only) in momentum space. I perform the Fourier transform, carry out the manipulations, then reverse the transform to get the answer. I believe this answer because my study of mathematics has convinced me that the manipulations I have performed are logically valid - they are truth-preserving. It is true that in mathematising the theory of quantum processes, we may at some point have abstracted away from physical reference, but we still have mathematical functions about which our statements can be true or false.

A truly formalist example would be if I had trained a neural network to produce reliable predictions of the behaviour of quantal particles. In this case the transformations performed have no semantic significance (though it might be possible to unearth them and back-fit them with an interpretation). However accurate the results might be, the uninterpreted neural network transformation does not guarantee the preservation of truth; success is only a matter of empirical likelihood, based on our experience of its performance to date. A pure formalism may be a heuristic means of generating interesting results, but it is not a proof based on reasoning from statements about the world.

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