Gottlob Frege: Numbers as Properties of Concepts

Firstly, let’s think in terms of predication – that most basic of logical procedures and an important element of nearly all traditional ontology.

Take this Frege-like statement:

Rihanna is one and Radiohead is three.

In terms of ontology, Gottlob Frege didn't think that numbers are properties of objects (i.e., objects like Rihanna and Radiohead). That is, we can't predicate “unity” or “oneness” of Rihanna in the way we can predicate “sexiness” of her. In terms of its grammatical form, that's mainly because the sentence above is a conjunctive identity statement. That is:

Rihanna = 1 & Radiohead = 3

According to extensional logic, it follows from this that if “Rihanna is one”, then “Beyoncé is one”. And if “Beyoncé is one”, then we can also write “Beyoncé = 1”, as before. However, according to the extensional principle of substitutivity, we now have:

Rihanna and Beyoncé are one.

or

Rihanna & Beyoncé = 1

That is, the proper names have the same reference – viz., the number 1.

Frege (after Kant) argues that this argument is also true of the predicate “exist” (or “exists”). This too can't be a predicate (or property) of a concrete object. We can say:

This man exists.

However, we "really mean" (or we must mean):

The concept [man] is instantiated.

In other words, the concept [man] has at least one instance. (Or, alternatively, there is at least one instance of the concept [man].) So if the predicate “exists” can only be applied to concepts (not to objects), then we can say that "existence" is “a predicate of predicates”. That is, a predicate of concepts, not of objects. We can also say that the predicate “exists” is a meta-predicate (or a meta-concept); which, unlike lower-order predicates, only applies to concepts. (Just as a meta-truths apply to truths about facts/observations/etc., but which aren't themselves about facts/observations/etc. - they're only linguistic expressions.)

More importantly for Frege, numbers are predicated of concepts. This is the case because, for example, the number 5 is the [class of all five-membered classes]. So, as before, we have a concept [5] which is applied to other concepts. This means that the number 5 is a meta-predicate; just like the predicate expression “exists”.

What about this statement? –

There are four politicians.

The Fregean “logical form” of that perfectly grammatical expression is:

The concept [politician] is instantiated four times.

That is, the predicate expression “politician” is used of four objects – i.e., four politicians. Again, in terms of logic, the concept [politician] itself is really predicated, not the actual concrete politicians. In consequence, only the concept [politician] is predicated with the concept or concept-number 4.

Does it now follow that in 4 isn't really a concept at all: it's a logical abstract object? Frege himself famously writes:

The concept [horse] is not a concept.

That seemingly paradoxical statement can be explained in the sense that in certain statements (including the one above) the concept itself is predicated, not the extension of the predicate (or concept). If that’s the case, it becomes the subject-term of the statement. Consequently, it must therefore be an object, not a concept (or a predicate). Hence the prima facie paradoxical nature of Frege’s statement about the concept [horse].

We can now say:

The concept [4] is not a concept.

The number 4 is (as it were) turned into an object: i.e., a non-spatiotemporal abstract object. Can we do the same with the predicate expression “exists”? That is, can we write the following? –

The concept [exists] is not a concept.

Is the “property” existence really an actual object – a thing of some kind? I don’t think that Frege did think of existence in the same way as he thought of a number. However, all the Fregean arguments seem equally applicable to the predicate expression “exists”; not just to the “four” in “There are four politicians’. And if Frege did think that existence isn't a genuine object like the number 4, then how did he argue for such a distinction?

Simon Blackburn on Philosophy

On Formalising Philosophy

Simon Blackburn comments on the formalising tendencies which began the tradition of analytic philosophy. He traces it back to Frege. This means, among other things, that the frequent arguments against “traditional metaphysics” certainly didn’t begin with the logical positivists. It began with Frege and no doubt existed even before that (as with Francisco Suárez and his repudiation of Aristotle). As Blackburn explains it:

“Many philosophers thought that they were on the verge of replacing ‘old-fashioned woolly metaphysics' with a rigorous, formalised philosophy that had as its core a logically perfect language, shorn of the vagaries or ordinary discourse. Once we had translated philosophical problems from ordinary language into this purified language of logic, the solutions to philosophical issues would follow as surely as night follows day. That optimism only lasted until roughly the Second World War.”

It's strange, then, that the “ordinary language philosophers” thought more or less the same thing as the philosophical formalists. Only this time the solution to “woolly metaphysics” was going back to ordinary language. That is, not only away from what the metaphysicians has said (or the way they put it), but also away from the formalisers (as epitomised by Frege, Russell and the early Wittgenstein). The ordinary language philosophers wanted to translate arcane metaphysics into ordinary language and in so doing sort out its problems or show them up to be of the metaphysicians’ own making (i.e., “pseudo problems”).

On the Mind-Language-World Triangle

Blackburn offers three approaches which philosophers since Hume have seen as been of primary importance to the whole of philosophy:

“You might say the first thing to do s to understand ourselves, as Hume does, for example; you might say the first thing you’ve got to do then is understand our language, and so do the philosophy of language; or you might think, no, what you’ve got to do is settle the nature of the world, the things that surround us and after that our own nature and the nature of language will fall out. That in a sense is the scientific approach – the rest is stamp collecting.”

So why not all these approaches at the same time? Or why not work with all three and see how they interact with each other (even if only at the peripheries)? Why the obsessive need to create a hierarchy (or pyramidical system) with one discipline at the bottom and all the rest on top of it?

It's also interesting that previous notions of First Philosophy aren't included in Blackburn’s list. Thus ontology was First Philosophy for Aristotle and others. Epistemology was First Philosophy for Descartes. Psychology (or “human nature”) was First Philosophy for Hume. The philosophy of language (in its logical guise) was First Philosophy for Frege. The scientific nature (or description) of the world was First Philosophy for the logical positivists and arguably for Quine and others. And from the 1960s onward, the philosophy of mind has more or less become First Philosophy for many philosophers - superseding or simply incorporating the philosophy of language.

One may conclude from all this that we should completely suspect the very idea of a First Philosophy and simply accept the interconnections between the philosophical disciplines. And it's one step on from this rejection of First Philosophy to take an “interdisciplinary approach” to philosophy and incorporate linguistics, computer science, cognitive science and other disciplines into its purview. This in itself can be seen as an acknowledgement of “philosophical holism” in whatever form it may take.

In summary, then, we have a “triangle” which includes psychology (or ourselves), the philosophy of language and the world itself. We needn't take any of these as being primary or fundamental.

On Philosophical Prose

Simon Blackburn has little time for those philosophers who glory in the complexity of their disciple or just in their own philosophical writings. He says that he thinks philosophy's “difficulties we compounded by a certain pride in its difficulty”. It's ironic, then, that some of the great philosophers were also good writers. Blackburn cites Russell, Ryle and Austin. I would also cite Plato, Hume, Quine, Putnam, Searle and particularly various American analytic philosophers - as opposed to English ones. (It's often the case that as the English philosophers are to American philosophers, so Continental philosophers are to English philosophers.)

Bad writing, technicality and sheer pretentiousness, however, shouldn't imply that all work on the minutia of philosophy should be shunned or limited. Of course not. Some papers are bound to be complex. Not necessarily because of the subject’s difficulty; but just because the issues and problems will be technical in nature and therefore have a high number of unknown technical terms. Indeed some technical terms will be needed and others may well be gratuitous – it depends on the philosopher concerned.

Blackburn makes some other interesting points about philosophical prose – at least in its bad guise. He quotes John Searle stating: “If you can’t say it clearly you don’t understand it yourself.”

So all the times I thought critically of myself for not understanding a particular philosopher’s prose, perhaps all along he didn’t understand his own prose; or, more importantly, he didn't understand the philosophical ideas he was trying – badly – to express. I assumed my own cognitive limitations or the damned complexity of the subject. However, perhaps all along it was just a case of the philosopher concerned being a bloody poor writer – regardless of the complexity of his ideas. Either that or he might well have been just plain pretentious! Certainly philosophers like this don’t follow the Quintilian dictum (as quoted by Blackburn): “Do not write so that you can be understood, but so that you cannot be misunderstood.”

Of course, literally speaking, if one writes “so that you cannot be misunderstood”, then one must also be writing “so that you can be understood” – the two approaches go together. However, Bernard Williams (also quoted by Blackburn) offered the obvious riposte to this “impossible ideal”:

“Williams snapped at that and said it was 'an impossible ideal. You can always be misunderstood', and of course he’s right. But I think the point of Quintilian’s remark isn’t 'write so as to avoid any possible misunderstanding’ but to remember that it’s difficult and that it’s your job to make it as easy as you can.”

It's interesting to note here that Williams’ impossible-ideal argument can also be used in favour of the idea that there will always be someone in one’s own culture - no matter how rational - who'll misinterpret at least something you write or say. Indeed perhaps everyone who reads or listens to you will misinterpret you in some small or large way. The idea of a perfect communication of a complete and perfect meaning to a perfect interpreter seems to be a ridiculous ideal. It seems to be almost – or even literally – impossible and for so many reasons. So you'll always be “misunderstood” by someone in some way. Indeed each person will misunderstand you in some way - whether that way is large or small. All we have left, as writers or philosophers, is to realise that “it’s [our] job to make it as easy as [we] can”. We can't be expected to do more than this. We can't guarantee the perfect communication of our ideas or the perfect understanding of our ideas by other people (as anyone who uses social media already knows). And even if we allow this slack, perhaps, in the end, it simply doesn't matter that much because communication doesn't require either determinate meanings or determinate interpretations. We seem to manage quite well in most situations without perfect languages and other philosophical ideals. It's only certain philosophers who get hot under the collar about the possibility (or actuality) of the problems of meaning or translation. You can't (to use Derrida's term) “mathematicise” meaning or interpretation/understanding. We now know, after all, that formalised or artificial languages have many severe limitations and we can even say that natural languages are far more expressive and pragmatically efficacious than even the sum of their alternatives.

On the Importance of Philosophy

Blackburn comments on the importance of philosophy and places it within the wider context of Western culture as a whole. He says:

“The high ground has got to be just that it’s one of the world’s great literatures. If you’re ignorant of Aristotle, Hume and Wittgenstein it’s like being ignorant of Shakespeare, Jane Austen or George Eliot and this ought to be regarded as shameful in the same way as ignorance of great literature would be.”

Not many would think of any philosopher’s work being great literature even if it's philosophically great. However, it's often difficult to disentangle the two - especially in the case of a philosopher like Plato. This is also the case if one accepts Jacques Derrida’s idea that there's no difference in kind between philosophy and literature: only a difference in degree. (I suppose this will partly depend on the philosopher we're talking about.)

For example, this would be easy to argue in the case of Plato (again). However, could we really see the work of Rudolf Carnap in the same way? We can also say that just as Shakespeare, Austin and Eliot have shaped the world, so too have Blackburn's philosophical examples – namely, Aristotle, Hume and Wittgenstein. I would add many other philosophers to the list of philosophers who've shaped Western culture: Plato, Aquinas, Descartes, Locke, Kant, Hegel, J.S. Mill and so on. However, I’m not sure if I would regard it as “shameful” if people hadn't read the philosophers on Blackburn’s list. I've never read much Montaigne or Plotinus and I've never read certain other well-known philosophers. (I suspect that Blackburn hasn't either.) However, if one hasn't ready any of these, then perhaps that is shameful and indeed quite remarkable.

Apart from (some) philosophy being “great literature”, Blackburn cites “pragmatic grounds” as to why philosophy is of vital importance – and has always been of vital importance. He writes:

“A much more interesting, pragmatic ground is that I really do think that unless people have some tools for reflecting on the language they use they’re apt to be behaving unselfconsciously, and unreflective behaviour is often behaviour that’s at the mercy of forces which we don’t understand. So I think that realising the state of your language is a very important device for realising he state of your culture at this time in history, and in politics.”

This isn't just a strong suggestion to study language in the philosophy-of-language sense: it's also a suggestion to study and critically analyse the words and concepts we use every day to see how they shape what it is we think; as well as the way they shape how we actually experience the world.

For a start, people will soon come to realise how our words and concepts not only reflect world and culture; but also determine or shape them. When we realise that, we can't help but reflect on the words and concepts we use and how we apply them. If we don’t do this, as Blackburn puts it, we'll be “at the mercy of forces which we don’t understand”.

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*) All the quotes in this piece are taken from the book, What Philosophers Think. In this book Julian Baggini and Jeremy Stangroom interview various philosophers and scientists.

Book Reviews (1): David Chalmers' *The Conscious Mind*

The Conscious Mind: In Search of a Fundamental Theory is the best book I’ve read on a single philosophical subject. Of course I may believe that simply because David Chalmers tackles subjects that I’m interested in and he does so in a way I appreciate. Nonetheless, there’s a fairly substantial consensus on this book — at least among those people who care about this issue and who’re part of the “analytic tradition”. 

For example, in 1996 the well-known American philosopher David Lewis wrote (some five years before he died in 2001) that The Conscious Mind “is exceptionally ambitious and exceptionally successful — the best book in philosophy of mind for many years”. Similarly, the British philosopher Colin McGinn said that the book is “one of the best discussions in existence, both as an advanced text and as an introduction to the issues”. And then Steven Pinker said that “The Conscious Mind is an outstanding contribution to our understanding of consciousness”.

My view is that The Conscious Mind: In Search of a Fundamental Theory has insights on virtually every page, is dense with argumentation and is clearly written; despite sometimes being technical. 

I just said that I believe it’s the best book on a philosophical subject for a long time. I say that even though I don’t agree with everything David Chalmers argues. Indeed I don’t even agree with most of what he argues. For example, I have serious problems with Chalmers’ very strong and frequent emphasis on logical possibility, zombies and intuition (which are all connected together by Chalmers). Still, Chalmers argues his case in a very strong manner; though obviously not strongly enough to convince me.

The Conscious Mind: In Search of a Fundamental Theory dates back to 1996. It was Chalmers’ first book; though he’d published academic papers before this (some of which date back to 1990).

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“I have advocated some counterintuitive views in this work. I resisted mind-body dualism for a long time, but I have now come to the point where I accept it… I can comfortably say that I think dualism is very likely true. I have raised the possibility of a kind of panpsychism. Like mind-body dualism, this is initially counterintuitive, but the counterintuitiveness disappears with time… on reflection it is not too crazy to be acceptable… If God forced me to bet my life on the truth or falsity of the doctrines I have advocated, I would be fairly confidently that experience is fundamental, and weakly that experience is ubiquitous.” — David Chalmers

I decided not to tackle any of David Chalmers’ topics in this review simply because the book is so dense with arguments. It just didn’t make sense to single out anything specific. And even if I had done, it would probably have turned this review into something else entirely.

Broadly speaking, Chalmers still holds most of the positions he articulated in this book. 

The Conscious Mind (as stated) is dense with argumentation. And partly because of that, Chalmers’ book fluctuates between reading like a paper in a technical philosophical journal (even if he steers away from soulless academese) and being a “popular philosophy” book. However, to be honest, though Chalmers’ writing is very clear, he rarely pulls off stuff that could be sensibly classed as “popular philosophy”. Indeed in the introduction Chalmers says that his “notional audience at all times has been [his] undergraduate self of ten years ago”. That’s not to say that there are no simple parts (or even simple chapters) in this book — there are. However, on the whole, it’s more technical than most “introductory” or popular books on philosophical subjects.

For example, the section ‘Supervenience and Explanation’ (which itself includes five chapters) is highly technical. Indeed one section seems like a convoluted detour into modal logic, possible-worlds theory and semantics. I suppose that Chalmers would see all this as being a necessary technical grounding for what comes later. Indeed in some of these chapters there’s hardly any discussion of consciousness. This is especially true of the long and technical chapter called ‘A posteriori necessity’ which is ten pages long and doesn’t contain a single mention of consciousness or the mind. The following twenty-four pages hardly mention consciousness either.

The most interesting chapters in the book (at least from a 2020 perspective) are ‘Naturalist Dualism’ and ‘Consciousness and Information: Some Speculation’ (which deals with panpsychism). That’s primarily because naturalistic dualism is peculiar to Chalmers himself and panpsychism has a lot of contemporary relevance. Many of the other chapters, on the other hand, have been done to death in analytic philosophy; specifically the stuff on qualia, phenomenal consciousness, the nature of reduction, etc. Having said that, since this book was written in 1996, perhaps these subjects hadn’t really been done to death at that precise moment in philosophical history.

The last chapter, ‘The Interpretation of Quantum Mechanics’, seems rather odd to me. It’s a strange add-on. It’s very difficult to see how Chalmers’ take on the various interpretations of quantum mechanics fits into the rest of the work. Here again consciousness is hardly mentioned. When it is mentioned, it’s in relation to how consciousness has been featured in the scientific tradition of quantum mechanics. Thus there’s stuff about observation and measurement. Consciousness also features more heavily when Chalmers covers Hugh Everett’s interpretation of quantum mechanics in which “superposition is extended all the way to the mind”. The idea of superposed minds is also tackled — and it’s all very strange!

I suppose that one reason that Chalmers writes twenty-five pages on the various interpretations of quantum mechanics is that the quantum mechanics-consciousness connection was becoming fashionable in the 1990s. However, it seems that Chalmers believed that most citations of quantum mechanics — when it came to consciousness — didn’t solve what he calls “the hard problem of consciousness”. And neither was he too sympathetic with the idea of “superposed minds” within the strict context of Hugh Everett’s “many-worlds interpretation” (which Chalmers believes is a misreading of the physicist’s theory).

The chapter ‘Consciousness and Information: Some Speculations’ is — obviously! — the most speculative. Especially the section on panpsychism. Indeed Chalmers happily admits that. He even says that “[t]he ideas in this chapter” are “most likely to be entirely wrong”. Whether or not Chalmers believe that now — some 25 years later — is hard to say. He’s certainly added much to his position on panpsychism; as well to his position on information theory.

Perhaps the chapters ‘Supervenience and Explanation’ and ‘The Irreducibility of Consciousness’ are the most important in The Conscious Mind. As stated earlier, there are also some technical (as well as somewhat tangential) sections in these chapters too. (The chapter on qualia is also detailed and technical.) It’s in these chapters that Chalmers articulates his most central and important point about consciousness: that it’s not reducible to the physical. It’s also here that he also states that “experience is a datum in its own right”. Therefore experience (or consciousness) needs to be treated that way.

Philosophy Now - "What are the moral limits to free speech?"

"What are the moral limits to free speech?

“Please give and justify your answer in less than 400 words.” - Philosophy Now (April/May 2018)

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Dear Editor,

It's odd really. Many people claim to be strongly in favour of free speech. Yet, as soon as you scratch the surface, you'll quickly find that almost all the people you talk to quickly realise (or acknowledge) that there must be at least some limits to free speech.

But there's a problem here.

People cite very different reasons as to why there should be limits to free speech. They also cite different examples of the kind of speech they believe should be limited (or made illegal). Having said that, it's also true that there are some well-known limits to free speech which almost everyone agrees upon. (Such as “shouting 'Fire!' in a crowded cinema” or encouraging paedophilia in public spaces.) Nevertheless, other proposed limits to free speech often tend to simply reflect people's extremely specific political biases. And because of that, it can be said that free speech would be drastically curtailed if all our political biases were acted upon by the state or by the legal system.

So perhaps any limits which are placed on free speech should be given a moral – i.e., not a political – justification. (Of course this is hinted at in the opening question.) Yet some people may now say that morality and politics are firmly intertwined when it comes to free speech! However, surely the two can be separated if the proposed limits on free speech are given abstract moral (as well as philosophical) justifications. In that way, even people who strongly disagree when it comes to politics could (at least in theory) accept such limits if they were given such moral justifications.

Despite all that, almost every moral justification of a limit to free speech will have its exceptions and opponents. It's also the case that extreme or perverse limitations on free speech could be morally justified. (Such as the argument that allowing people to debate race or violence will inevitably encourage racism or violence.) Self-referentially speaking, even limiting (or banning) the public discussion of the question “What are the moral limits to free speech?” could be morally justified.

Surely this must mean that no single moral justification of the limits of free speech will ever receive universal approval or acceptance. Nonetheless, a complete consensus may not be required in the first place. After all, no philosophical, moral or political justification or position will ever please everyone. And that, of course, isn't necessarily a bad thing.

Yours,

Paul Austin Murphy.

My Letter to Philosophy Now - 'Heidegger's Ways of Being'

Dear Editor,

In the Philosophy Now piece, 'Heidegger's Ways of Being', Andrew Royle claims that Martin Heidegger offered us a “direct refutation of Rene Descartes' solitary introspection”. Is that really the case?

Descartes “global scepticism” was an epistemological exercise. It had little - or nothing - to do with ontology. It was about how Descartes – or about how we - could know, and then philosophically demonstrate, that (to use Andrew Royle's own words) “the world and other people actually exist”. It wasn't even that Descartes didn't believe that the world and other people existed. Descartes' enterprise was about his knowledge of other people's existence. Indeed Descartes' initial scepticism is also what's called “methodological scepticism” (or “methodological doubt”). That is, it was supposed to be sure route to knowledge. It was a philosophical method which was designed to show us that knowledge of the world and other people is possible.

As for the Heideggerian grammar of the word 'I'.

Say, for argument's sake, that the use of the word 'I' also (as Royle puts it) “necessarily refers to... 'you' or an 'other'”. How did Descartes know that all the people he'd experienced weren't also the simulations of an “evil demon”? Thus such simulations (or mental distortions) might have also grounded Descartes' use of the word 'I'.

To put that another way. If the Matrix and “brain-in-a-vat” (Hilary Putnam) scenarios are possible, then it's equally possible that the simulations we have of other people may ground our use of the word 'I'. Indeed one can even say that a Heideggerian notion of “social Being” (or Dasein) can exist alongside Cartesian scepticism – indeed even if the Matrix and brain-in-a-vat scenarios are possible. (Putnam, of course, argues that his own scenario isn't possible – and for loosely Heideggerian reasons!)

As for solipsism. To quote Arthur Royle himself:

“Although Heidegger's argument works to abate Descartes' solipsism... Whilst the 'I' (or 'ego') was indubitably alone for Descartes...”

In everyday-life terms, Descartes would have left his doubts well behind after he'd solved (or thought he'd solved) the “sceptical problem”. (Just as Hume forgot his own scepticism when playing billiards.)

This means that Descartes most certainly wasn't a solipsist. (Though it can of course be said that he was a “methodological solipsist” for the duration of the Cogito.)

A genuine solipsist is someone who does indeed have an ontological position on what Royle calls the “I” or “ego”. What's more, a solipsist feels the reality of his solipsism throughout his life. (Or at least he does so for as long as he's a solipsist or thinks about his philosophical predicament.) Descartes, on the other hand, took a journey from his radical scepticism to a sure knowledge (or so he believed) of the world and other people. Now that's very far from being solipsism.

Not only that: solipsism has ontological and ethical implications. However, that isn't really the case when it comes to Descartes' scepticism. Having said that, it's indeed the case that certain political and sociological theorists have interpreted Descartes' scepticism as a 17th-century philosophical expression of “bourgeois individualism”. Yet even if that were true, Descartes never made this explicit. With Heidegger and solipsists, on the other hand, their ontological and ethical positions are indeed made explicit.

Yours,
Paul Austin Murphy.

James Ladyman on Structural Realism

In his Understanding Philosophy of Science), James Ladyman says that “structural realism” was “introduced” by the philosopher John Worrall. 

This position - within the philosophy of science (though mainly within the philosophy of physics) - has it that structures are fundamental. What's more, structures are real (hence the word “realism”).

At the heart of structural realism is the idea that physics essentially deals with structures, not with “things” or entities. More importantly, it is these structures which are retained in physics; not the things which physics posits. That is, such structures can be passed on from an old theory to a new theory (i.e., when both theories are - ostensibly - about the same phenomenon or problem).

Thus structural realism is a realism about structure, not about things, conditions or “empirical content”. As Ladyman puts Worrall's position:

“... we should not accept full blown scientific realism, which asserts that the nature of things is correctly described by the metaphysical and physical content of our best theories. Rather, we should adopt the structural realist emphasis on the mathematical or structural content of our theories.”

More relevantly

“Since there is (says Worrall) retention of structure across theory change, structuralism realism both (a) avoids the force of the pessimistic meta-induction (by not committing us to beliefs in the theory's description of the furniture of the world), and (b) does not make the success of science... seem miraculous...”

Thus structural realism has an argument against the well-known position of “pessimistic meta-induction”: i.e., such pessimism only applies to “empirical content” and “theory”, not to structure. That is, structure is retained “across theory change” and thus total inductive pessimism is unjustified.

So three questions arise here:

i) What is structure?

ii) Is structure really retained “across theory change”?

iii) And (this is related to i) above) how is structure distinguished from “empirical content” and “theory”?

Henri Poincaré's Structuralism

James Ladyman traces structural realism back to Max Plank and Henri Poincaré. For example, Ladyman quotes Plank stating the following:

“... 'Thus the physical world has become progressively more and more abstract; purely formal mathematical operations play a growing part.'...”

Indeed Plank's position is one which most physicists would uphold; even if they wouldn't use Plank's precise wording. Of course this is simply an indirect acknowledgment that that there would be no physics without “mathematical operations”; or, more broadly, without mathematics itself.

Thus, in a broad sense, the structural realism position is that it's all about the maths. Or, at the least, it's all about the mathematical structures noted by physicists.

Ladyman also tells us that Poincaré

“talks of the redundant theories of the past capture the 'true relations' between the 'real objects which Nature will hide for ever from our eyes'...”

So here Poincaré's response to the pessimistic meta-induction is to argue that “true relations” are retained from theory to theory; even if the things or phenomena mentioned in the theories aren't. As can also be seen, Poincaré uses different jargon to contemporary structural realists in that he talked of “true relations” rather than “structures”. Then again, structural realists also make extensive us of the word “relations”. After all, it's the structures of the physical world which account for these relations; and, in a sense, they're also constituted by such relations.

Nonetheless, Poincaré does seem to depart from contemporary structural realism when he talked about “real objects”. As can be shown, structural realists (especially ontic structural realists) dispense entirely with objects or things - “real” or otherwise. Or at least they believe that “every thingmust go”. Despite that, since Poincaré qualifies his reference to “real objects” with the clause that “Nature will hide [these real objects] for ever from our eyes”, then it can be said that Poincaré – effectively - did indeed dispense with objects/things too. That is, when Poincaré used the phrase “for ever from our eyes”, presumably he wasn't only talking about literal visual (or observational) contact with real objects. He must have also meant any kind of contact with them – including (as it were) theoretical contact. Thus Poincaré's real objects were little more than Kantian noumena and therefore of little use in physics. Then again, since Poincaré was also a Kantian, noumena might well have had a role to play in his metaphysics and physics.

So was Poincare a Kantian and a structural realist at one and the same time?

Just as Poincaré used the words “true relations” instead of the word “structure”, so the philosopher of science Howard Stein uses the word “Forms”. That is, Stein says (as quoted by Ladyman) that

“our science comes closest to comprehending 'the real', not in its account of 'substances' and their kinds, but in its account of the 'Forms' which phenomena 'imitate' (for 'Forms' read 'theoretical structures', for 'imitate', 'are represented by'”.

Clearly Stein is attempting to tie contemporary structural realism to a long philosophical - and indeed Platonic - tradition. He does so with his use of the words “Forms” (with a platonic 'H') and “imitate”. Then again, he also rejects the equally venerable (i.e., in the history of philosophy) “substances” and “kinds”.

Having said that, this very same passage can be read as expressing the position that Forms (or “theoretical structures”) are actually imitating (or “representing”) “substances and their kinds”. So, as with Kantian noumena, it's not that substances don't exist: it's that our only access to them is through theoretical structures: i.e., through the mathematical structures and models of physics. If this reading of Stein is correct, then that makes his position almost eliminativist. As with Kant's noumena, Poincaré's “real objects” and ontic structural realism's “things”, aren't Stein's “substances” also (to use Wittgenstein's words) “idle wheels in the mechanism”? What purpose do they serve? Do they serve as a abstract Kantian “grounding” or as a Lockean “I know not what”? Or, to quote Wittgenstein again, perhaps it's best to conclude: “Whereof one cannot speak thereof one must be silent.”

Examples from Physics

Maxwell and Fresnel

James Ladyman cites John Worrall's example of the structural elements of Augustin-Jean Fresnel's theory (of light waves) passing over to James Clerk Maxwell's later theory. Ladyman quotes Worrall thus:

“... 'There was an important element of continuity in the shift from Fresnel to Maxwell.'... ”

More relevantly, this

“'was much more than a simple question of carrying over the successful empirical content into the new theory'...”

However, neither was it just a case of “carrying over or the full theoretical content or full theoretical mechanisms”.

Thus, if it's not just a case of “empirical content” and “theoretical content” being “carried over”, then what else was also carried over? The answer to this is: structure. That is, Fresnel's theory shares a certain structure with Maxwell's later theory. Or as Worrall himself puts it:

“... 'There was continuity or accumulation in the shift, but the continuity is one of form or structure, not of content.'...”

Clearly Worrall doesn't see Fresnel's and Maxwell's theories as only being (what's often called) “empirically equivalent”. He states that it's not (only) “empirical contents” which are passed on. That must mean that the two theories are also theoretically “under-determined” by the empirical content.

This means that Fresnel's and Maxwell's theories are neither empirically nor theoretically identical. So does that mean that these two theories are structurally (or formerly) identical instead? Worrall may not also believe in complete structural identity between these two (or any) separate theories. However, he clearly does believe that structural identity is more important (or more substantive) than any empirical or theoretical identity.

Thus it follows from all this that we'll now need to know how it is, precisely, that structure is distinguished form both empirical and theoretical content when it comes to the theories of Fresnel and Maxwell – and indeed when it comes to any comparative theories in physics.

Newton & Quantum Mechanics

Worrall also attributes a structuralist position (if not an explicit acceptance of structuralism) to Issac Newton. Worrall describes Newton's structuralist reality (if not his position) in the following manner:

“... 'On the structural realist view, what Newton really discovered are the relationships between phenomena expressed in the mathematical equations of his theory.'... ”

In certain respects, this is certainly true. For example, it's often and justifiably stated that quantum mechanics wouldn't so much as exist without its mathematical descriptions and predictions. John Horgan, for one, states that

“mathematics helps physicists definite what is otherwise undefinable. A quark is a purely mathematical construct. It has no meaning apart from its mathematical definition. The properties of quarks – charm, colour, strangeness – are mathematical properties that have no analogue in the macroscopic world we inhabit.”

Isn't all the above just as true of much of Newton's work? However, it's certainly the case that Newton wasn't an eliminativist when it came to things/objects (or when it came to Poincaré's “real objects”). Despite that, it was still “mathematical equations” which captured the things or phenomena Newton was accounting for in his theories.

The question (as with quantum mechanics) is:

Is there any remainder after the mathematics (or mathematical structure) is taken away?

What's left? Kantian noumena or, well, literally nothing? Of course it's hard to defend an eliminativist position when it comes to Newton and the concrete things he was talking about (e.g., stars, the moon, etc.). However, eliminativism seems much more appealing and justified when it comes to the micro world of quantum mechanics. In this realm, everything really does seem to be mathematical. Quite simply, there are no genuine equivalents to the moon, stars and even gravity (at least Newtonian gravity) in the quantum realm. In other words, our only access to the micro world is through mathematics. Clearly, that can't also be said of the world as described by Newton.