By Leonard Susskind, What Is the Universe? February 2026
Wednesday, April 22, 2026
3682. Book Review: Michael Roberts on Three Books about Capitalist History and Transformation, Controlling or Replacing Capitalism
By Michael Roberts, Michael Roberts Blog, April 22, 2026
This post reviews some recent economics books published by various authors, both Marxist and non-Marxist.
Let me start with a magnum opus, Capitalism – a global history, by Sven Beckert. Beckert is the Laird Bell Professor of History at Harvard University, where he teaches the history of the United States in the 19th century and global history. His ‘Capitalism’ is called a ‘monumental book’ by global inequality expert, Thomas Piketty, himself an author of an eariler gargantuan publication back in 2014 called Capital in the 21st century (Piketty’s suggestion then was that he was ‘updating’ Marx’s Capital from the 19th century).
Beckert in contrast is not trying to update or critique Marx’s Capital. Instead, as an economic historian, he aims to paint a broad canvas of the rise of capitalism from its early embryonic origins, that he takes back to 1000 years ago. He does not provide a theoretical analysis of capitalism as Piketty tries in his book. This book is very much more descriptive than analytical. He delivers a global view of capitalism , not confined what he calls the ‘eurocentric’ approach of others. That is the book’s merit, full of anecdotes and examples of capitalists at work worldwide. But the book’s de-merit is its lack of any systematic understanding of capitalism. Indeed, it is like the work of Adam Tooze – namely, it is ‘more the how, than the why’.
As the blurb says for the book, “Sven Beckert, author of the Bancroft Prize–winning Empire of Cotton , places the story of capitalism within the largest conceivable geographical and historical framework, tracing its history during the past millennium and across the world. An epic achievement, his book takes us into merchant businesses in Aden and car factories in Turin, onto the terrifyingly violent sugar plantations in Barbados, and within the world of women workers in textile factories in today’s Cambodia”.
Capitalism, argues Beckert, was born global. Emerging from trading communities across Asia, Africa, and Europe. And capitalism can only be described as a global phenomenon. “This book understands capitalism as, above all, a global development whose local articulations can only be understood globally. The economic dynamics of a given place are inescapably shaped by its connections to the outside world. There is no “French capitalism” or “American capitalism”; rather, there is capitalism in France and America, which have contested and complicated relationships with capitalism elsewhere, indeed everywhere “
Beckert makes big claims for the revolutionary nature of capitalism. “It was a fundamental break in human history not just because it revolutionized economic affairs but because it turned human relations upside down; infiltrated our politics, societies, and cultures; altered the natural environment we inhabit; and made revolution a permanent feature of economic life. The capitalist revolution is the only revolution whose fundamental core is that it is ongoing, that it qualifies as a state of permanent revolution.”
But of course, he recognises that capitalism has its faults. “Capitalism is also distinctive for the particular kinds of social inequalities and global hierarchies it creates.” But Beckert does not want to take sides between those authors supporting and those critiquing capitalism. “On one side, Marx’s writings became sacred texts through which to filter the politics du jour; on the other side, scholars read capitalism’s history through the equally sacralizing lens of Adam Smith’s writings. This book strives to avoid either idolatrous extreme.”
Actually, it is not true that Marx did not recognise the great changes that capitalism made to human progress; or that Adam Smith saw no faultlines in market economies. But Beckert resorts to descriptive history rather than economic insight. As Beckert puts it: “this work is an effort to reclaim capitalism as a territory for historical investigation. This history will show that capitalism is neither a state of nature nor a process whose internal logic determines its eventual outcome in more than the most general way.” So the Marxist materialist conception of history and Marx’s explanation of the internal contradictions in capitalism are to be put aside; as are the views of mainstream neoclassical economists that markets and profit making are an eternal and beneficial feature of human social organisation. Instead, capitalism is a contingent history.
Beckert does not hide the brutal nature of the emergence of capitalism globally. “Although capitalism’s history is often told as a story of contracts, private property, and wage labor—that is, stylized as a history of the realization of human freedom—there is another story, equally important, about vast expropriations, huge mobilizations of coerced labor, brutality in factories and on plantations, fierce destructions of noncapitalist economies, and massive extractions of resources for private gain. Capitalism rested, as we will see in the chapters that follow, not just on productivity gains but on enormous appropriations”.
Many of the early sections of the book give the reader a panoramic view of the capitalist process at work across the world, even when other social formations like, slavery, feudalism and Asian despotism were dominant. Unfortunately, when Beckert gets into 20th century, the period when capitalism became fully dominant globally as the mode of production and social formation, Beckert’s analysis becomes weaker. He notes the post-1970s crisis of reconstructed capitalism, ie the neoliberal period, but it seems he remains confident that capitalism is here to stay despite the accumulating economic, environmental and geopolitical crises that we see accelerating in the 21st century. ”We can anticipate that capitalism will remain a global totality, even if the nature of that totality continues to change, perhaps in radical and surprising ways. We can expect capitalism’s enormous creativity to persist, along with its amazing adaptability.”
Or does he? “Eventually, however, there will be a moment when capitalism ends. Regardless of whether we fear or hope for that end, capitalism, like everything in human history, is finite, even if it is impossible to say when or how it will end or what will replace it.” But even if capitalism is to give way to a new stage of human social organisation, it will take a very long time and “be interwoven within capitalism itself, just as capitalism was itself embedded in noncapitalist societies for centuries.” Or maybe not – if the “ecological and social crises unfolding right now and right here become unbearable”. All these maybes are a product of his descriptive approach to the history of capitalism.
Another opus magnum is the latest book by former World Bank lead economist and global inequality expert, Branco Milanovic. I have posted several times on Milanovic’s indepth studies of global inequality, but this new book is not so much about inequality but more about what he considers is the great transformation in the world economy that is taking place –namely the movement of economic power from North America and Europe to Asia. “The first defining change is the much greater importance of, and the movement of economic activity towards, Asia and the Pacific.”
The second big change is the result of that shift. As China became richer, the Chinese people also became richer. That meant that people who were in the lower-middle class in the US, Germany, or Italy, for the first time in the last 200 years, fell behind substantial numbers of people from Asia. “At the level of the nation-state, we have had a movement towards much greater importance of Asia in economics and politics. At the level of personal incomes, we see the decline of the Western middle class.”
Milanovic argues that the Industrial Revolution transformed the countries that were leading the industrialization—the UK, France, Northern Europe, then the United States, and finally Japan—and made their people much richer than people elsewhere. But In the last 40 years, we have had, for the first time, a serious challenge to that. Countries in Asia are now not only catching up but, in some cases, even overtaking Western countries technologically.
This has led to a new cold war not now based on ideology (capitalism v communism, as with the US and the Soviet Union), but now economically between the US and China. If China continues with real GDP growth rates 2-3% points higher than the US rate, within one generation, and a maximum of two generations, you will have the same number of people in China who are above the US median income as Americans. “If one thinks that the real sign of catching up is when China becomes equally rich on a per capita basis as the United States, it will take a long time. But before that happens, China as a nation would be much more powerful than the United States simply because it is so much bigger.” But see my forthcoming paper on Catching up , to be published by the World Association of Political Economy.
Milanovic says there are three views on the benefits or otherwise of the globalisation of trade and finance in the last 40 years. The mainstream one is that trade among nations benefits all countries and so leads to peace. Adam Smith is more nuanced and argued that only ‘balanced trade’ would maintain peace. But there is the Hobson-Luxemburg-Lenin theory, which holds that the big powers would fight for control of the resources and assets of the rest of the world and that would eventually lead them to war ie imperialism. Milanovic tends to a mix of the last two views. The end of globalisation and free trade has led to a loss of living standards for many in the West and thus “a huge dissonance between different parts of the Western population.” I would add that globalisation led to a massive transfer of value and resources from the Global South to the Global North, hitting living standards not just in the Global North but also for the vast majority in the Global South..
According to Milanovic, neoliberal globalism has now been replaced by ‘national market liberalism.’ Tariffs are being imposed and immigration controls are increasing. The world has moved from option two to option three. ”We still have neoliberalism, but only at the national level. We end up with a version of neoliberalism stripped of its international component.” Milanovic concludes that “we clearly have a global disorder.” But he lays his hope on the world moving towards a multipolar system. Eventually, “we can build a more equitable international system where major powers have a greater stake than they do now.” So a new balance of trade and finance and economic power can emerge. Option three becomes option two again, hmm.
Mariana Mazzucato is another rock star economist of the ‘left’, once called the world’s scariest economist. I have reviewed many of her previous books (search my blog). But it seems she does not really scare the international powers that be. She is regularly invited to speak around the world at various mainstream economic gatherings and as an adviser to governments. Her latest book is called The Common Good Economy. This follows on from a previous book, the Mission economy. – each time a new attractive title suggesting economic innovation and insight.
Mazzucato tells us that “Our economic system is broken. The climate crisis is accelerating. Inequality is deepening. Public trust is crumbling. Wealth concentrates in fewer hands while governments scramble to fix what markets can’t do, rather than to shape them from the outset.” So what should well-meaning governments do? Instead of trying to correct these ‘market failures’ and trying to patch up problems, governments need “to proactively build the economy we need”. She offers a ‘new theory of the common good, one which allows governments and businesses to develop purposeful economic relationships, creating value and building spaces where human flourishing can happen.”
Mazzucato continues her merry way across the globe at conferences, government meetings etc to advocate ‘mission projects’; conditionalities on big business and a social contract between workers and bosses – all for the ‘common good’ economy. Dare I say it, but clever jargon and trendy titles do not make for radical change.
Ann Pettifor in her new book, Global Casino, does not even look for radical change. You see, unregulated global finance is causing the crises we see in the world economy. The global market in money – housed in the offshore ‘shadow’ banking system – holds $217 trillion in financial assets and operates beyond the reach of any nation’s taxman. Asset managers, private equity firms, and pension and sovereign wealth funds scoop up the world’s savings for investment and manage them as they choose, unaccountable to politicians or the citizens who elect them.
But it does not require socialist or very radical measures to sort this out. Pettifor: “societies and governments can take back control of the global financial system. We have done it before and can do it again. Indeed, it is imperative that we do so, if we are to manage the twin threats of climate breakdown and biosphere collapse.” Pettifor reckons that in those halcyon days after the second world war, a global financial order was established with the Bretton Woods agreement to manage ‘global imbalances’ and currency and trade flows as well as regulation of financial excesses and recklessness. But President Nixon blew all this up internationally when he took the dollar off the gold standard in the early 1970s and later government leaders deregulated the financial sector, turning the world economy into a giant casino. This was the reason for the global financial crash in 2008-9 – it has nothing to do with falling profitability of capital or any other rigid Marxist explanation. The answer now is to return to the post-war period of managed trade and financial regulation – simples. But I think not.
You see Pettifor knows what she is talking about – unlike the rest of us on the left. “What always strikes me about the great financial crisis of 2007–9 was that the Left didn’t know it was coming. I am very proud of having written The Coming First World Debt Crisis (2006), but the rest of the Left didn’t see it coming. People talked about globalization as if it was a given. And then when it blew up, there was no plan B. We didn’t even know it could happen. We were as stupid as the chair of the Federal Reserve, Alan Greenspan. The Left was as stupid as Greenspan, who said he didn’t believe it could happen.” Actually many on the left (at least the Marxist left) did see the financial crash coming (see my paper here). And what is this plan B to replace globalisation and unregulated speculative finance? According to Pettifor, it is restoring proper regulation. But regulation always fails. Indeed, since the Great Recession, there have been several banking crises, despite increased regulation.
Moreover, if the cause of all our woes globally is an uncontrolled financial sector, why does Pettifor not call for the public ownership of the banking system in the major economies and the closure of hedge funds and other speculative forms of finance capital? Instead, Pettifor offers a tax on speculative financial transactions and capital controls on footloose capital flows – and which governments are going to introduce these? This is like putting a bandage on a gaping wound with blood flowing from a pierced artery.
Mazzucato offers us capitalism with ‘conditionalities’ for the common good and Pettifor offers us capitalism ‘regulated and managed’. Only one book proposes ending the capitalist mode of production and it is not by a feted academic, but by an Irish Marxist activist. James O-Toole’s Economics for the Exploited is written from a working class point of view. He explains clearly and simply how capitalism works and why it cannot deliver the needs of humanity any more.
O’Toole covers Marx’s law of value and answers its critics clearly (he explains Marx’s law of profitability and even deals with the so-called ‘transformation problem’). He explains the cause of economic crises, inflation and the rise of imperialism. And he outlines the case for a planned economy under common ownership and democratic control as the way forward for humanity and the planet.
“Modern humans have been on Earth for around 300,000 years. Class society is a few thousands years old and capitalism only a few hundred. There’s nothing “natural” about this system. In those few hundred years capitalism has brought us to the point where corporate greed could actually destroy the natural underpinnings of any advanced social order. The clock is ticking. This system isn’t natural. We can live in other ways. We workers produce this system. It’s in our hands. Workers have to take control.”
Friday, April 10, 2026
3681. John Mearsheimer: Trump's Only Option Is Surrender
By John Mearsheimer, Breaking Points, April 10, 2026
Thursday, April 2, 2026
Wednesday, April 1, 2026
3679. Are Mathematical Objects Real?
By Stephen Ferguson, Philosophy Now, 1997
I work on the philosophy of mathematics – in fact, I’m just finishing off my PhD dissertation. When I meet people and they ask what I do, they always look puzzled by my reply. Typical responses include surprise that mathematics has a philosophy, complete bewilderment, or else they conclude that I must do some form of esoteric mathematics and confess that they could never get their head round maths when they were at school. The situation is usually slightly better when I meet other philosophers – they know that it has something to do with Bertrand Russell; sometimes they will mention Wittgenstein or even Frege or Gödel. But no-one ever seems to be able to put their finger on just what it is that I do.
In a nutshell, the philosophy of mathematics deals with the special problems that arise from our possession of mathematical knowledge. Therefore it is a branch of epistemology, the study of how we know things, just as philosophy of science and philosophy of perception are. Unlike other forms of knowledge, where we learn by experience, mathematical knowledge seems to be purely concerned with the realm of thought. In addition to specific questions about mathematics, discussion also concerns how mathematical knowledge fits into the broader scheme of things, and more general accounts of our cognitive capacities.
Truth is generally considered to be a matter of correspondence between our thoughts and language on the one hand and reality on the other. As mathematical statements are (hopefully) true, this suggests that the objects referred to in these statements – numbers, sets, functions, etc. – must exist. In the past twenty years, two questions generally attributed to American philosopher Paul Benacerraf have become paramount. They are:
1) given that mathematical objects don’t have causes or effects, how can we refer to them?
2) given that mathematical objects don’t have causes or effects, how do we have any knowledge of them?
These have been called Benacerraf’s twin puzzles of Referential and Epistemic Access. As you can imagine, there has been much debate about different ways to explain these puzzles.
In what follows, I would like to outline some of the main responses to these puzzles, to pass on some of the feel for the philosophy of mathematics, after which I’d also like to outline the position that I advocate myself.
I. Mathematical Realism
There are three, not entirely separate, main lines of response to these puzzles, which are usually adopted depending upon whether the philosopher thinks that there really are numbers, in the same way as there are tables, chairs and beermugs – this is what is called a realist or platonist response. Alternatively, they might deny that there really are numbers, and explain our apparent commitment to them as a convenient fiction, claiming roughly that mathematics is not about some abstract realm of mathematical objects, but about certain concepts which we employ: this might be called an anti-realist strategy. Thirdly, there are those who argue that to concentrate upon individual objects – such as natural numbers or sets – is the cause of the problems; instead we should be concerned with mathematical structures. Typically structuralists also espouse some form of realism about structures, but this is not always the case.
There are three main arguments for mathematical realism. The first, put forward by German mathematician Gottlob Frege (1848- 1925) relies on the reality of language. He argued that where our language does genuinely engage with the world, singular terms (what he called Proper Names) refer to, or stand for, objects. So the proper name ‘chair’ refers to an actual object. He suggested that true statements such as those of numerical identity (e.g. 2+2=4) supply the requisite contexts to conclude that numerals too refer to objects, i.e. that numbers are objects. This is sometimes called linguistic or semantic realism, because of the prominence of language in this account.
The second argument is due to Kurt Gödel, the logician who had the office next door to Einstein at the Institute of Advanced Studies at Princeton. Gödel argued that mathematicians are able to perceive mathematical objects using a special faculty of intuition. This is Platonism proper (with a capital ‘p’) because, as with Plato’s theory of the forms, an abstract realm is postulated as well as a means of being acquainted with that realm.
Thirdly, Hilary Putnam – one of America’s most influential philosophers today – has put forward an argument based on Quine’s arguments for epistemological holism. Quine contends that our knowledge is a unity, and that it is not possible to isolate one aspect of our knowledge from all others. Putnam’s argument runs like this – if we take our scientific theories seriously, we are committed to believing in the entities postulated by those theories, even when it isn’t possible to experimentally verify their existence. However, if our knowledge really is holistic, as Quine claims, then the theoretical entities aren’t the only objects to which we commit ourselves by accepting a theory: as mathematics is involved in expressing physical theory, we are also committed to accepting the existence of mathematical objects. This is the Indispensability Argument – so called because mathematics is indispensable in science.
II. Anti-realism about mathematics
Traditionally two positions oppose platonism: Intuitionism and Formalism. Recently a variety of new anti-realist positions have also been articulated, notably Quasi-realism and Irrealism. More of them later.
Intuitionism was first put forward as a philosophical account of mathematics by Dutch mathematician Jan Brouwer (1881-1966) as an alternative to Platonism. According to him, mathematics is not about an abstract realm of mind-independent objects, but rather about the creation of mathematical objects by the human mind. Rather than a statement being true or false of some preexisting mathematical reality, he argued that we create that reality as we go. Mathematical statements are true or false when they deal with objects already created, but neither true nor false when they break new ground. So what logicians call the Law of Excluded Middle – that for every statement, either it or its negation is true – must be rejected, if Brouwer is correct. But as the proofs of certain results in ‘classical’ mathematics rest essentially on the use of this Law, denying it requires us to rethink much of the subject.
Over the past 30 years, Michael Dummett, sometime professor of logic at Oxford, has argued for Intuitionism by attacking Frege’s semantic arguments for realism. On the basis of the Law of Excluded Middle, classical mathematics is committed to the existence of statements which are true, but which we cannot prove are true; these are known as evidence transcendent truths. A good example is Goldbach’s conjecture that every even number is the sum of two primes. Most mathematicians are reasonably certain that this is true, but as proving it would involve finding the prime components of infinitely many even numbers, such a proof could never be completed. Dummett’s arguments concern how we learn mathematics; someone has to teach us, and moreover, we have to be able to show that we’ve understood what we’ve been taught. He argues that if knowing the meaning of a statement is knowing what the terms within it stand for, as Frege claimed, then we would never be able to learn the meaning of evidence transcendent statements, as no-one could show us the truth of them. This has become known as semantic anti-realism, and like the Intuitionism that inspired it, if correct, will involve the rejection of the Law of Excluded Middle.
Like Intuitionism, Formalism opposes the Realist conception of truth which supports Platonism. The usual interpretation of Formalism is that it treats mathematics as being fictional or like a game; but this would be a misinterpretation of at least one Formalist – the most famous of all: David Hilbert (1862-1943). His attempts to deflate the metaphysical worries about truth might be put in modern terms by saying that there is nothing more to truth than is entailed by a principle of correspondence so weak as to be a platitude:
‘P’ is true if and only if P
More recent attacks on Realism have come in the form of Quasi-realism and Irrealism. Quasi-realism is a term coined by Simon Blackburn, in connection with debates in the philosophy of language and in ethics. He suggests that in certain discourses, although the surface grammar has a certain shape, it misrepresents the underlying logical structure of the syntax. This enables us to carry on our various linguistic practices, without being bogged down with burdensome philosophical commitments – such as to the existence of non-physical objects. Geoff Hellman has used this strategy to suggest that mathematical statements are not straightforward indicative claims, but instead that they are subjunctive claims about what the consequences would be if there were such objects as numbers or sets.
Quasi-realists accept that mathematical statements are true, but deny that there are any numbers, sets or functions corresponding to the names occurring in the surface grammar. lrrealists on the other hand, such as Hartry Field, have argued that mathematical language should be taken at face value, but as there are no such objects as those named in the statements of mathematical theories, these statements must be false. Field takes his task as two-fold: to show that what he takes to be the main argument for mathematical realism – the Indispensability Argument – is misguided, and to show that despite mathematical statements being false, they can nevertheless be useful. He accounts for the usefulness of mathematics by demonstrating that all mathematics satisfies a certain principle of normativity which he calls conservativeness; Inputting true information into the mathematical machinery will result in output which is also true. He argues that this principle of conservativeness shows that mathematics is ultimately no more than a convenient shortcut, and that science can be conducted without express mention of mathematics.
III. Structuralism
Unlike most of the mathematics studied by professional mathematicians, the surface grammar of arithmetical practice suggests that particular objects are at stake. But when it comes to the items studied by working mathematicians, it is not objects but structures which are of prime importance. One typical mathematical structure is the group : given a set g and a binary operation +, ⟨g, +⟩ is a group if the following hold:
(G-i) closure – for any a, b in the set, there is some c in the set such that a+b=c;
(G-ii) identity – there is an element, e, so that for any element a, a+e=e+a=a;
(G-iii) inverses – each a element has another element, b, related to it, so that a+b=b+a=e. This is the inverse of a, written a-1;
(G-iv) associativity – for any a, b, c in the set (a+b)+c=a+(b+c).
Rather than think of any individual object in the structure as having an important mathematical role, the key insight when dealing with structures is that the whole structure is mathematically important: no part of it can perform in isolation. Perhaps the best way to grasp the mathematical concept of structure is to consider physical structures. In his forthcoming book, Stewart Shapiro describes structures as being similar to sports teams. Take a typical football team: there is a goalkeeper, players in central defence, midfield and with the strikers up front. Some teams may play with three strikers, others with two: these then have a different structure.
Now consider the problems that faced platonism: like numbers, positions in a football team are not physical things, so how do we have knowledge of them? The obvious answer is that we come to know what role a goalkeeper plays by watching several instances of goalkeepers, and abstracting what is common to all of them. We would be missing something if we thought that all goalkeepers had brown hair: what is important is the role that they play within the structure – within the team. The structuralist argues that if all of mathematics is considered structurally (not just the obvious algebraic structures such as groups, but also areas such as arithmetic) then this perspective offers simple solutions to various philosophical worries, such as Benacerraf’s twin puzzles.
We learn mathematics by spotting patterns – by what is called pattern recognition. In fact, Resnik argues that we would be better off talking generally in terms of patterns than structures, but the differences is merely terminological. Both Resnik and Shapiro, two of the most influential structuralists, have argued that mathematical knowledge comes by first experiencing various ‘concrete’ or physical patterns, and then abstracting to the underlying structure.
I’ve already briefly mentioned Geoff Hellman’s modal structuralism, as an example of a quasi-realist strategy; like Shapiro and Resnik, he takes the philosophical problems with platonism to stem from its focus on mathematical objects, and suggests that a structurebased account would remedy this. Unlike Shapiro and Resnik however, he bases his structuralism not on pattern recognition, but on the thought that structures describe the possible combination of objects, and so takes statements of mathematics to be concerned with our grasp of what is possible and what is necessary.
Structuralism is attractive because it offers an account of the subject matter of most of modern mathematics – unlike platonism which focuses almost exclusively upon the mathematics with which the man in the street is familiar. However, the exact shape of the structuralist’s arguments are worth considering in closer detail. First there is this obvious desire to account for mathematical practice. Secondly, the claim that ‘All mathematics is structural’ is made for strategic rather than philosophical reasons. By this I mean that it gives the structuralist certain advantages. For example, without this claim, the structuralist’s theories would only apply to the obviously structural areas of mathematics, such as the theory of groups mentioned above. In order for structuralism to be an alternative to platonism, the structuralist needs to show that this view also applies to ordinary mathematics, i.e. to arithmetic.
The usual way to do this is to show the adequacy of the structural view of number. Rather than think of the natural numbers as being a collection of individual objects, it is possible to think of them as a structure, with each element of the structure being the successor of another element of the structure. Nothing can be said about any of these elements, except for the relationship it holds with the others. What distinguishes such structures from those of algebra, is that the underlying pattern of the natural numbers is categorical – for any given cardinality, all the models of the structure are the same. In the algebraic case, as with football teams, there may be different models of the structure, exhibiting differences similar to the differences between one team with three strikers, and another having only two.
IV. Modest structuralism
I’ve never been convinced that the structuralist is correct about arithmetic – I’ve always thought that the intuitive differences between, on the one hand, systems such as the natural numbers, real and complex numbers, and on the other, structures such as groups, was more than just a matter of the number of models of a theory.
I do think that the structuralist is basically correct about abstract algebra – which has led me to try to produce an account which retains all of the intuitive appeal of Frege’s platonist account of ordinary mathematics, along with the structuralist’s account of professional mathematics. Recall that Frege offered linguistic arguments for his conception of numbers as objects: to offer a two-fold account such as I’ve just proposed, entails giving reasons based on the features of mathematical language, to suggest why structures and systems might differ.
As well as considering arithmetic, Frege also wrote about the properties of ordinary language. He argued that in most sentences, terms which stood for the same object could be exchanged without changing the truth value of the sentence. For example, if it is true that:
(A) Lois Lane loves Clark Kent
then it is also true that:
(B) Lois Lane loves Superman.
However, in certain contexts, this breaks down. We all know that for the longest time, Lois was unaware of Superman’s secret identity, so it is false to say that:
(C) Lois Lane believes that Superman is Clark Kent
but true to say that:
(D) Lois Lane believes that Superman is Superman.
Frege concluded that in non-indicative statements, proper names do not in fact stand for the objects they usually refer to, but instead proper names refer to the way in which we usually work out what the reference is, i.e. what he called the sense of the name.
Like statements of belief, statements of modality also cause problems for reference. Recall that Hellman argued that structures deal with the possible combinations of objects; this suggests that mathematical statements are not straightforward indicative claims, but instead that they are subjunctive claims about the consequences were there such objects as numbers or sets.
It is possible to combine these two approaches: I claim that statements of arithmetic are indicative propositions, and so the proper names that they contain make essential reference to objects: so numbers are objects. But statements of structural mathematics are not indicative – they are subjunctive statements about the consequences were there objects arranged in such-andsuch a fashion and so do not refer in the same way that arithmetical statements do.
V. Conclusions
I’ve tried to convey some of the main issues in the philosophy of mathematics, and hopefully have managed to make it appear accessible and interesting. Just about everything on the bibliography below is readable, although it varies in technical difficulty; for example, Shapiro’s first book is full of logical detail, while his second book contains virtually no complicated logic at all.
I have hinted that the philosophy of mathematics deals with whether there really are numbers, sets and functions. Different positions such as platonism, intuitionism and formalism offer different ways of tackling these questions; structuralism on the other hand, offers a radically different approach, providing a new perspective on the debates.
Rather than accept the structuralist slogan ‘All mathematics is structural’, I prefer to think of there being structural and non-structural areas of mathematics, and give separate though not independent accounts of each area, based on the differences in the context of statements of those discourses.
Bibliography
Benacerraf, P (1965) ‘What numbers could not be’, Philosophical Review 74, pp47-73; (1973) ‘Mathematical Truth’, Journal of Philosophy 70, pp661-80 – Putnam, H (1983) Philosophy of Mathematics: Selected Readings 2nd edition, Cambridge University Press
Blackburn, S (1984) Spreading the word. Oxford: Clarendon Press
Brouwer, J (1949) ‘Consciousness, Philosophy and Mathematics’, in Benacerraf & Putnam (1983), pp90-6
Dummett, M (1973) ‘The philosophical basis of Intuitionistic Logic’, in Benacerraf & Putnam (1983), pp97-130
Field, H (1980) Science without numbers. Oxford: Blackwell
Frege, G (1879) Die Grundlagen der Arithmetik; trans. Austin (1950) as Foundations of Arithmetic. Oxford:Blackwell; (1893) Die Grundgesetze der Arithmetik. Vol I, Olms: Hildesheim
Gödel, K (1947) ‘What is Cantor’s Continuum Problem?’, American Mathematical Monthly 54, pp515-25, reprinted in Benacerraf & Putnam (1983), pp470-86
Hellman, G (1989) Mathematics without number. Oxford; Clarendon Press
Putnam, H (1971) Philosophy of Logic. New York: Harper
Resnik, M (1981) ‘Mathematics as a science of patterns: Ontology and Reference’, Noûs 15, pp529-49; (1982),‘Mathematics as a science of patterns- Epistemology’, Noûs 16, pp95-105
Shapiro, S (1991) Foundations without foundationalism. Oxford Logic Guides 17, Oxford University Press; (forthcoming) Philosophy of Mathematics: Structure and Ontology. Oxford University Press
Wright, C (1983) Frege’s conception of numbers as objects. Scots Philosophical Monographs, Aberdeen University Press
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