Nancy Cartwright on Scientific Laws

Cartwright on Non-Modalised and Non-Universalised Scientific Laws

Nancy Cartwright doesn’t believe that there are ‘modalised universal or probabilistic generalisations’ (201):

I think there are no descriptive laws in the sense of modalised universal or probabilistic generalisations – there are none of those which are not ceteris paribus. I don’t think it follows from that there are no true nomological type statements in physics. When you ascribe a capacity to an electron, because of its charge, to repel other electrons or attract protons, I think that capacity-ascription is universal and it’s true. But I think it in no way can be translated into a modalised universal generalisation. It has to be seen as a capacity statement which works in a very different way than one of these laws as traditionally conceived. (201)

I think it is a very radical position to say that there are no "modalised universal or probabilistic generalisations – there are none of those which are not ceteris paribus" (201) in physics or any of the other natural sciences. That means that no law of physics is necessary or necessarily true. Of course "probabilistic generalisations" by definition can't be necessary or necessarily true. All these generalisations (or laws) will contain ceteris paribus "escape clauses" which will say that they only hold if so-and-so or this-and-that is the case or if "all things remain equal". This must mean that all things never remain equal. That is, there are always some ceteris paribus amendments of some kind thus the generalisation (or law) can never be "modalised". 

However, Cartwright goes on to say something which appears, prima facie, to go against what she says in the first sentence. She says that there are "true nomological type statements in physics" (201). That must mean that ‘nomological’ is not a modal term. That is, something can be "law-like" (or even a law) without it being necessary. We can have laws of physics without them being necessary or even universal in nature. This seems to go against not only scientific and philosophical conceptions of natural laws, but even what the layperson believes.

Cartwright, instead, talks in terms of the "capacities" of natural phenomena like, in her example, electrons. This may well be a "dispositional" analysis of causation or of natural phenomena. If it is about the capacity of, say, an electron to do this or that, then this analysis must also be counterfactual in nature in that it isn't telling us what the electron is doing, but what it would do given such-and-such circumstances. More specifically, the electron has the capacity (or contains the disposition) "to repel other electrons or attract protons" (201). Again, this doesn't mean that a particular electron (or even all electrons) is repelling other electrons or attracting protons, only that it would do so given such-and-such a situation. 

Is this necessarily the case? Is it highly probable or probable at all? However, despite the fact that Cartwright said that that there are no "modalised universal or probabilistic generalisations", she now says that the "capacity-ascription is universal and it’s true" (201). This appears to go against her earlier statement, as just expressed. However, perhaps the universality of the capacity-ascription is itself counterfactual in nature in that it isn't universal as it stands; though it would be universal given such-and-such a situation or set of conditions. This may simply mean that at no single moment in time is every electron either repelling other electrons or attracting protons. However, each and every one of them would do so given such-and-such conditions. That is, even though a single (or even all) electrons aren't at this moment repelling other electrons or attracting protons they would do given such-and-such conditions – that is, perhaps, if they were contiguous with other electrons or with protons.

So it's no surprise that Cartwright still says that this capacity-ascription can "in no way be translated into a modalised universal generalisation" (201). Because of this dispositional analysis of electrons, we can say that because it's the case the every electron isn't repelling other electrons or attracting protons at this or any moment in time, then we can't state universal generalisations about electrons or say that each or every electron must be necessarily behaving in such-and-such a way. It may even be the case that no electron in the universe is either repelling other electrons or attracting protons. This will certainly be the case when it comes to individual electrons and their behaviour. It's not surprising, therefore, that this seemingly counterfactual analysis of laws or of electrons "works in a very different way than one of these laws as traditionally conceived".

In conclusion, this account of causation and natural law seems to tally very well with David Hume’s empiricist account. (Hume is still seen to be of vital importance when it comes to these issues in the philosophy of science and even in science itself.)

To conclude. Cartwright still accepts the nomological reality of natural laws and scientific generalisations and even concludes herself by saying that "the planetary system is a nomological machine". That is, everything about the planetary system is law-like and works according to the laws of physics; though none of these laws are modal or necessary in nature; and we can’t offer any true universal generalisations about them unless they contain ceteris paribus amendments which, of course, effectively render them non-universal or counterfactual in nature.

Cartwright on Capacity-Ascriptions

Nancy Cartwright goes into more detail as to why she offers a dispositional (or capacity-ascription) counterfactual analysis of scientific law or causation:

"I think most cases of causation are cases of interaction and that they’re not intelligible in a scientific way. That is, not much of what happens in the natural world is governed in a systematic way, that it takes an enormous amount of effort in special background conditions in special circumstances before you get regular repeatable behaviour. The more standard view is that everything that happens is an instance of some regularity, albeit a very complicated or a very abstract one which we may never know. And since I think it’s very difficult to get regularities at all, they’re not just lying around and everything that happens is an instance of them, then I genuinely believe that most things that happen in the world can’t be subsumed under a regularity, or ought to be subsumed under a regularity. A lot of what happens simply is a result of interaction which we can’t have a handle on." (202)

This dispositional (or counterfactual) analysis really does go against the standard account of causation and scientific law. We still have causation (or regularity); though "most cases of causation are cases of interaction and that they’re not intelligible in a scientific way". So instead of talking about 'causation’ at all perhaps we should talk about simple ‘interaction’, as Cartwright does. Another way of putting this is to say, as Cartwright does, that "not much of what happens in the natural world is governed in a systematic way". This too goes against traditional conceptions of science. Not only against the Newtonian idea of the systematicity of nature; but even of its very intelligibility. That is, if one doesn't have systematicity (or even regularity), then this will make the natural world a lot more difficult to understand because the application of laws will be all the more limited and complex (if not particular). In order to get regularity and systematicity 

"it takes an enormous amount of effort in special background conditions in special circumstances before you get regular repeatable behaviour" (202). 

If one does in the end ‘find’ regularity and the instantiation of natural law, they come at the cost of much analysis of "special background conditions in special circumstances". That is, the ceteris paribus list will be large and complex. And if some cases of regularity are "very complicated or very abstract one which we may never know", one can apply anti-realist arguments to such a fact by saying that an unknowable regularity (one that is even unknowable in principle) isn't an example or a case of regularity at all! That is, if we can't fathom a regularity, then how then do we know that it must be a regularity in the first place if we have no evidence for it or no means of knowing that it's in fact some kind of regularity? Cartwright doesn't say that "most things" can't be subsumed under a natural law. She says that most things can't even "be subsumed under a regularity". Not only that: it's not even the case, in all instances, that they "ought to be subsumed under a regularity". Perhaps subsuming literally everything under a regularity is a kind of forcing of the issue in order to make scientific investigation (or research) simpler and more amenable to explanation.

Finally, Cartwright uses the word ‘interaction’; though not the word ‘regularity’. However, if we don’t have regularity, then how can we have a natural law? Or is it really the case that we need regularity even in the cases where we do have instantiations of or subsumptions under a natural law? Perhaps regularity - let alone necessary and universal regularity - simply needn't go along with the notion of a scientific natural law. Perhaps we can have lawhood in one and only one case or instance (as in Toovey’s ‘singularist’ account of causation?).

It's because of all this that Cartwright ops for a counterfactual or capacity-ascription account of scientific knowledge. That is, natural phenomena have counterfactual capacities (or dispositions) to do such-and-such in such-and-such situations and conditions (given enough ceteris paribus variables or laws). It’s just that there's no universal necessity - or sometimes even regularities - involved in these capacity-ascriptions of natural phenomena. All we have is the situation in which we can say: 

X would - or may - do A, B and C, given situations or conditions D, E, and F.

However, it's not necessary (or even highly probable) that X will do A, B and C. And it certainly isn’t the case that it is doing A, B and C.

Cartwright gives her own example of a "singular causal claim" thus:

"Acorns have a capacity to give rise to oak trees. Now, I think that’s true and I think that it goes hand-in-hand with a lot of singular causal claims: like ‘That oak tree in my garden came from the acorn we planted there twenty years ago.’ And there are a lot of true singular causal claims like that... without there being any regularity about what would repeatedly happen in any of the circumstances in which an oak tree did result." (202)

What seems to being said here is that all we can have (at least in many cases) are "singular causal statements: we can't have explanatory regularities. However, this doesn't seem to rule out the possibility (or actuality) of lawhood. Is this because of this singular causal claim

"That oak tree in my garden came from the acorn we planted there twenty years ago."

contains ceteris paribus laws or specific conditions for the causal process that is being described or explained? That is, it's a reference to a specific acorn and a specific oak tree - not to all acorns and all oak trees. Not only that: there's also a reference to a specific garden and a specific act of planting an acorn twenty years ago. Thus, unlike most (or all) scientific generalisations, this is a tensed statement - not a timeless one (not an "eternal sentence", as Quine put it). In that case, how can it be a scientific statement at all if it's so particular and specific? Perhaps instantiations of lawhood (or nomological causation) which are also scientific in nature simply don't need to be universal or modal in nature. In other words, not all acorns - by necessity - must grow into oak trees. Perhaps the type of garden or soil must also enter into the equation. Indeed perhaps things were different twenty years ago in that acorns required conditions that they no longer require. Or the nature of soil may have substantially changed or changed just a little. However, on a counterfactual or capacity-ascription analysis, given the required set of initial conditions and factors, acorns will grow into oak trees. It’s just that these initial conditions or factors are too numerous or variable to be subsumed "under a regularity" or perhaps even under a natural law (or a set thereof).

Cartwright then offers us what she takes to be the "standard view" of these issues:

"You know, the standard view is – take those circumstances and if you had a good microscope and had sufficient time, in every one of those cases in which the singular causal claim is true, you could find some description of the circumstances for that very acorn such that, if only you could repeat those circumstances, there’d be a universal generalisation. Now, I think basically that’s all just a metaphysical pipedream." (202)

What's being said here amounts to a truism. That is, if you had the same circumstances in another case, then the same singular causal claim would be true of that identical set of circumstances and conditions. It follows that if this were repeated for every acorn, then indeed we would have a "universal generalisation". The problem is that these specific circumstances and conditions aren't likely to be exactly replicated by another set of circumstances and conditions. 

This amounts to the claims:

If x, y and z were the case, then x, y and z would be the case. 

Or more generously:

If x, y and z bring about A, B and C,
then A, B and C would also be the case.

There's always the practical problem with finding these universal sets of conditions and circumstances in that one would need "sufficient time" and the requisite equipment to account for them. One would need an infinite amount of time to account for the sets of conditions and circumstances that would be required to hold down a universal generalisation about all acorns and their growing into oak trees. So not only is this both practically impossible and an indefinitely (or infinitely) complex task, Cartwright also calls it "a metaphysical pipedream" perhaps born of the metaphysicians' desire to bring order out of complexity and simplicity out of multiplicity. 

In anti-realist terms, even if the metaphysical pipedream were true about acorns and oak trees under a set of conditions and circumstances, then what the metaphysician says is the case could not be known to be the case. We would simply not have the requite evidence (or data) to substantiate or confirm (let alone prove) his ostensible universal generalisation about these or any other phenomena in the natural world. In that case, it's indeed a metaphysical pipedream and perhaps says more about the metaphysician’s desire for – or dream of – order and simplicity than it does about the true nature of the natural world.

Despite everything that's been said, Cartwright doesn’t have a problem with what she calls "causal happenings". Indeed why should she? She only rejects modal notions and unjustifiable universal generalisations. This, at least in these respects, is a thoroughly Humean account of causation and natural law. Again, what she stresses instead of these traditional notions of causation and law are singular causal claims:

"So causal happenings are as much part of the fundamental ontology of the world as anything else and then it’s very hard to construct a reason why there can’t be causal patterns – acorns tend to give rise to oak trees and that’s borne out in a lot of singular causal happenings – without there having to be universal generalisations in the background." (203)

So, just like Hume, Cartwright isn't against the reality of causation per say; but against the "projection" (as Simon Blackburn puts it) of modal properties -such as necessity and universality - onto all acts of causation. Not only that: even without necessity and universality, "causal happenings are as much part of the fundamental ontology of the world as anything else". Perhaps it's simply the case that we don't need modal notions or universality in science and perhaps not in philosophy either. Either that or there simply are no modal properties such as necessity and universality and thus such things are never justifiably projectable or applicable to the natural world. Necessity and universality are just metaphysical notions which metaphysicians stick onto the world and then take to be real ontological properties. In the end they are nothing but projections onto the world or psychological ideas (as Hume might have put it).

Finally, Cartwright has no problem with "causal patterns" either, just as long as there are no "universal generalisations in the background". Singular causal events (or processes) may instantiate causal patterns (as it were); thought they do so without also instantiating a universal generalisation or being subsumed under such a thing. 

Cartwright gives a simple and common example of such a causal pattern. She says "acorns tend to give rise to oak trees". It’s as simple as that really. And that singular causal statement (or statement about a set of singular causal statements) needn't assume (or bring on board with it) anything to do with universal generalisations or any modal notions of any kind. All we require (perhaps even in science) are singular causal statements such as the one just given.

We can conclude with the basic Humean claim that there simply is no such thing as logical necessity (or even metaphysical necessity) in the natural world. Just about everything that Cartwright has said so far follows from that fundamental Humean premise about causation and natural law. Just as we can never infer anything about the future from the conditions of the present; so we can’t even infer anything about another part or aspect of reality now from a given part or aspect of reality – at least not with logical or even a metaphysical certainty born of logical or even metaphysical necessity. Or as Cartwright puts it:

"Besides, I don’t think that this whole story makes sense, that there’s some description of every circumstance, and if you just went through the whole catalogue of all the properties that obtain on the occasion you’d finally find exactly that arrangement of them that would give rise to repeatability." (203)

Here again we have the formula: 

If conditions x, y and z, then A, B and C will happen. x, y and z are the case.Therefore A, B and C will be the case.

It's of course likely to be the case if one has 

"some description of every circumstance, and if you just went through the whole catalogue of all properties that obtain on the occasion"

then of course 

"you’d finally find exactly that arrangement of them that would give rise to repeatability". 

The problem is that an account of every circumstance - and all the properties which obtain for a singular causal relation or connection - will rarely (if ever) be forthcoming. And even if such a thing were possible, perhaps that wouldn’t help much in terms of scientific description and explanation. Of course if the same circumstances and properties reoccur, then the same effects will occur. But is this likely? And even if it is, would that in itself get us anywhere?

Populations: Arithmetical & Geometrical/Exponential Increase

You can partly get to grips with exponential increase by thinking about what is called The Malthusian Catastrophe.


Thomas Malthus claimed that the world's population is growing geometrically; though food production is only growing arithmetically.

What did he mean? In terms of food production, he was talking about the rate of new acres open to agriculture each year. He believed that the rate was fixed. Thus the food supply works, for example, like this: 100, 102, 1004, 1006.... 1012.... etc. That is, the acreage grew by two acres per year in Malthus's' day. It was a fixed rate of increase which never changed.


On the other hand, populations don't work that way. Basically, the more adults who can have children, the more babies there will be. However, that rate isn't fixed as in 2, 4, 6, 8... etc. This is shown, or at least it was, in terms of the entire population of the world. Instead of 2, 4, 6, 8... etc., it was more like : 1, 2, 4, 8, 16, 32... Instead of the simple addition of 2, we have a number being doubled every time. Thus if you compare 2, 4, 6, 8, 10 with 2, 4, 8, 16, 32, we have a difference of 22 even though both progressions included only four changes. The geometrical increase ends in 10, whereas the geometrical increase in 32. Both rates included only four changes or progressions. Clearly, as the progressions increase, the gap between the arithmetical ratio and the geometrical ration will keep on widening. Or, in Malthus’s case, the population increase will keep on outstripping the increase in food production – resulting in starvation, etc.

Another word for geometrical increase is exponential increase. Exponential increase is applicable to most or all living organisms. It's also applicable to human populations. Another way of putting this is to say that exponential increase is proportion to the given number being increased. With arithmetical increase, it is just the addition of 2 (or 3, 7, etc.) each time. That isn't the case with exponential increase. The larger the number, the larger the increase.


Many arguments against Malthus’s argument have been advanced. That is, even if populations increase, it isn't a necessary, or mathematical, fact that more people are likely to starve. However, even if Malthus’s prophesies were false, which they were (in the UK at least), the exponential increase of families is still (largely) true. Or at least it's true given other (many other) conditions, such as: that all off-spring themselves have families and that what happened to the first family, will happen to all further families generated by that first family. In other words, all the women need to be fertile, all the children need to survive into adulthood, and all the children need to get married, etc. If all these factors occur, then there will indeed be an exponential increase in the population number.


We can say that if all the conditions remained the same for Malthus's arithmetically increased food production, he would have been right.... But that's just it – he didn't foresee the possible other conditions in either the geometrical or the arithmetical case.

However, you can have exponential increase that isn't precise or which fluctuates. (Does that automatically stop it from being exponential increase?) That is, in theory, a couple could have six kids. Those six kids may get married and each have six kids. That would amount to 36 people in two generations. And if those 36 people did the same thing, that is, each have six kids, then the number of people would now be 36 times 6, which is 216 people. So from one couple, and in three generations, 216 people have been produced! That's an increase from 2 to 216 in twenty or so years. Now that's just one family that has produced 216 people in just twenty or thirty years. What about ten or a hundred families?


Take 100 families which have each produced 216 persons in twenty years. That's 216 times 100, which is 21600 people in twenty years. That is, 100 families have produced 21600 people in twenty years.

Hypotheses & Observational Consequences

How are Hypotheses Tested?

According to Wesley Salmon, test their hypotheses by deducing observational consequences from them. This suggests a degree of independence of the hypothesis from observations or from testing/experiments, etc. Indeed it suggests, quite simply, that the hypothesis comes before all observations or tests. An isn't that how the layperson, after all, takes the word hypothesis? He would say that the whole point of a hypothesis is that it's a stab in the dark.


Yet if a hypothesis, scientific or otherwise, were really a complete stab in the dark, what would be the point of it? It would be arbitrary as well as a complete fabrication. It would be plucked from the air and would therefore, surely, have no relevance to anything scientific. For example, I can formulate the hypothesis that the sun is made of cheese. What's the point of it? It's clearly false and clearly scientifically illiterate. Nonetheless, the suggestion seems to be that this is what a hypotheses is – a complete stab in the dark.

Thus we can conclude one thing: even though an hypothesis comes before its observational consequences, it clearly doesn't come before all previous observations. It clearly doesn't come before a whole lot more as well: previous theories, laws, tests, experiences, etc. To use the language of old-style epistemology: an hypotheses is not aprioristic. If it were truly a priori, in the traditional rationalist sense of that term, then it wouldn't be a scientific hypothesis (though I suppose it could be a non-scientific hypothesis).


Anyway, the hypothesis is tested in the sense that the observations it predicts, or the observational consequences of its content, do come to pass.


Now we meet two more scientific terms: confirmation and disconfirmation.


If the predicted observations or events occur, then hypothesis is confirmed. If they don't, it is disconfirmed.


I mentioned that no hypothesis can be truly free-standing. Another reason for this is that the formulation of the hypothesis must assume various or many things. These assumed things have been called auxiliary hypotheses because they underpin the given hypothesis. Or, alternatively, the auxiliary hypotheses are contained within the (new) hypothesis.


Wesley Salmon gives the example of a medical experimenter who predicts that a certain bacillus will be found in the blood of a certain organism. Now in order to be scientifically sure or certain of his experiment or hypothesis, he must accept certain auxiliary hypotheses about the optics which are part of the experimental set-up and which includes the microscope itself. However, in actual fact, the scientist in practice doesn't really need to assume or even accept any hypotheses about optics at this - or any - moment in time. He can ignore them. He's not a scientist of optics nor an expert on microscopes. Other scientists are. What this really means is that both logically and scientifically these auxiliary hypotheses underpin his experiment even if the medical experimenter need not - and probably will not – know a thing about these extra hypotheses: he hasn't got the time.

This is an example of Quine's scientific holism and the scientist concerned need not be aware of the “web of science” or even large parts of it. The web exists regardless of the particular knowledge of the individual scientist. It existed before him, during the experiment and it will exist after the experiment.







Deductive Consequences?

It may seem strange to argue that “a true observational consequence follows deductively from a given hypothesis”. Or, more precisely, it is the use of the word “deductively” that may seem strange. Surely deduction is a purely logical matter? How can anything “observational” deductively follow from, well, anything logical or even from anything non-logical? Surely only theorems, conclusions, etc. can deductively follow from something or even from a given hypothesis. That is an understandable position.


Nonetheless, if the hypothesis has a certain given content, and that content says that if H then O, then if H then O it doesn't matter if the content of O includes predictions about observations, experience or that which is empirical. After all, this is in fact a conditional statement. That is, if this hypothesis is correct, then O will occur. It doesn't matter if the hypothesis, or O, has empirical content, or says that something observational will occur. It says that if it were correct, then there would be certain observational consequences. The hypothesis, or conditional, generates, as it were, what deductively follows from it, even if what deductively follows from it are indeed observational consequences. In other words, the hypothesis is not claiming that there are logically deductive sequences in nature, as it were. It's saying that given hypothesis H, then O (the observations consequences) will follow. The deductive relationship is between H and O, not between one aspect of the world and another (in a non-Humean manner, as it were). Alternatively, there is no necessity in the world, but there is a certain kind of necessity, or at least a deductive consequence, from H to O.



Hypothesis and Evidence

An argument about the independence of an hypothesis from its evidence (or from evidence generally), or from observational consequences, can be articulated by saying that given exactly the same evidence, various and many hypotheses can explain that same evidence. Basically, this is a way of making the obvious point that evidence and hypothesis are not the same thing. Alternatively, an hypothesis is more than the evidence which supports it. (This is basically a rephrasing of the idea that “theory is always underdetermined by all available and relevant evidence”.)


In fact, just as I stated the truism that hypothesis and all available and relevant evidence are not the same thing, and also that rival hypotheses can explain the same evidence, so too it is the case that all these rival hypothesis - which are fighting to explain the same evidence - are not equal or identical either. And that lack of identity or equality, again, has nothing to do with the relevant evidence (which is the same for all the rival hypotheses).


It is commonly said, by both scientists and philosophers of science (though less by the latter), that these other factors include the degree of simplicity of the hypothesis as well as its explanatory power, esthetic value, comprehensiveness, etc. However, even though I have stressed the fact that evidence is not everything, it is, obviously, of vital importance. (How could it not be in science?) So it is true that many commentators, not always scientists, have stressed that Watson and Crick were esthetically delighted by the beauty of the double helix hypothesis for the structure of the DNA molecule. That's true. However, if they had wanted purely artistic pleasure they would have become painters or composers. That esthetic pleasure was largely generated by the simplicity of the double helix hypothesis. But simplicity, in scientific theory, is not an end in itself. That simplicity, in this and in many other cases, meant that there was a good chance that the said hypothesis is true/correct. That is, simplicity generated beauty and that beauty/simplicity generated the strong possibility of a correct hypothesis.

The Paradox of the Barber Who Shaves Everyone Who Doesn't Shave Himself

There are actually some (as it were) false paradoxes: arguments or situations which seem paradoxical until they are seen not to be paradoxical – simply false. Some logicians claim that this is one.



Well, does this barber shave himself? He must do because he shaves all those who don't shave themselves. But if he shaves himself, he can't be a member of [the class of those who don't shave themselves]. Yet his job is to shave all those who don't shave themselves – and he only becomes a member of that class when it's seen by himself that he doesn't shave himself.

Thus we have a contradiction. As a person who doesn't shave himself, he must shave himself. If he did shave himself, then he wouldn't be a member of [the Class of Those Who Don't Shave Themselves]. But the resultant situation, in both cases, is that he shaves himself. If he shaves himself, then obviously he shaves himself. However, if he doesn't shave himself, and he must shave all those who don't shave themselves, he must also end up shaving himself.

Why on earth should we assume that there isn't a barber who shaves all and only those who don't shave themselves? It's certainly not illogical if he excludes himself. But perhaps that's precisely the problem – he can't exclude himself! However, even if he can't exclude himself, the paradox doesn't appear to disappear: talk about not assuming that there could be a barber who only shaves those who can't shave themselves seems to be to sidestep it the problem.



The philosopher Roy A. I believe that Sorensen makes the mistake of saying “we should not assume that it is possible for there to be a barber who shaves all and only those he does not shave”. That locution doesn't appear to make sense. It's not paradoxical – just senseless.

The end result of this possible paradox is that we have a barber who shaves all and only those he does not shave. But I still can't work out how you get there. Therefore is may well be a mistake – perhaps just a typing mistake. The thing is, I don't know!

So perhaps Sorensen's locution is correct after all. The end result of this possible paradox is that we have a barber who shaves all and only those he doesn't shave. However, I still can't work out how Sorensen gets there. Therefore may well be a mistake – perhaps just a typing mistake.

David Hilbert

David Hilbert embarked on an enormous enterprise: the reduction of the whole of mathematics to a set of axiomatic systems. Clearly these systems must have been interrelated in many ways in order to determine and guarantee their mutual consistency. We can distinguish each axiomatic system by their different axioms. In addition, in order to move from these axioms to theorems, we must utilise something that is shared with logic: the rules of inference. Again like logic, one such rule of inference is the well-known modus ponens. This shared interest in the rules of inference is partly accounted for by the fact that "they are properties of reasoning as such". That is, both logic and mathematics must use the inferences which belong to every rational mind and even every thought or act of reasoning. They were, therefore, as foundational and fundamental as Aristotle’s ‘laws of thought’.

However, there must be some things, other than numbers, which distinguish mathematics from logic. For example, a prime candidate for this difference is the nature of the axioms in mathematics. They are different because they describe space, time and measurement in all its forms. These applications, to space, time and measurement clearly distinguish mathematics from logic, at least from pure formal logic but not, for example, quantificational logic.

Hilbert was a Platonist. He didn’t reject numbers in his systems. We cannot "eliminate the idea of number from the axioms". They are, then, fundamental to maths precisely because they are used in its many axioms. Hilbert expressed his Platonism in an even more platonistic way. Like Plato himself, "we must therefore suppose that numerical expressions stand for objects, which have a reality independent of our calculations". He was clearly not, therefore, a mathematical constructivist or a mathematical Wittgensteinian, and neither was he a mathematical Kantian. In addition, although these number-objects are "known to us through proof, but which are entities over and above the proofs by which we discover them". So Hilbert accepts that they are only known to us through proof, which is an operation and perhaps a psychological operation, these number-objects do not need us in order to exist. The numbers can then be called evidence-transcendent, at least in the case of those numbers and operations that can never be known, or are not known now. They are also mind-independent objects, even though they can be ‘known’ by minds.

In some cases, then, some object-numbers and operations may never be known as a matter of necessity. Take ‘Golbach’s theorem’ which cannot be proved (see Kripke’s 1971). These positions seem very counterintuitive to many minds, not only non-philosophical minds. Indeed they seem even stranger in the context of that branch of mathematical constructivism known as intuitionism. In the case of intuitionists, if a mathematical statement has not been proved or disproved it is, in fact, neither true nor false, mind-independently or otherwise. In fact, unproven statements have another ‘truth-value’: indeterminate. Incidentally, this is also the case with ‘future contingents’ – statements about the future. They too are neither false nor true, but they are indeterminate instead. Perhaps this is because a future-statement cannot be proved either, almost by definition.

In addition, being constructivists, intuitionists do not believe that numbers are objects either, whether abstract or concrete. We ‘construct’ numbers by the operations we carry out on them. They are not found, either, via platonic ‘intuition’ or Husserl’s ‘direct insight’. If there were no minds, there would be no numbers and no mathematical operations on these numbers.