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Left: L. E. J. Brouwer (1881–1966). Right: Michael Dummett (1925–2011)

Mathematical intuitionism (which is a subset of mathematical constructivism) has it (or had it) that mathematics is purely the result of the mental activities of human beings; not the discovery of mathematical entities which exist in an “objective” (or Platonic) realm.

In terms of L. E. J. Brouwer’s original intuitionism (i.e., of the first three decades of the 20th century). Brouwer believed that a mathematical statement is a subjective claim which is verified in terms of the validity of a particular mental “construction”; which is, in turn, dependent on human intuition.

As for anti-realism

Anti-realism is an epistemological position which was first explicitly articulated by Michael Dummett in the early 1960s. Taken more broadly, various (weak, strong or vague) forms of anti-realism (though not the term itself) can be said to date back to the ancient Greeks. Indeed it can even be convincingly argued that the Copenhagen interpretation of quantum mechanics is anti-realist. …


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There are various philosophers, logicians and scientists who’ve argued that Nature (or the world) is fuzzy, inconsistent and/or uncertain.

It’s hard to grasp what the words “fuzzy”, “inconsistent” and “uncertain” could even mean when applied to Nature (or to any given x in the world). This is the case even of things “at the tiniest scales”. It’s easy, on the other hand, to accept that things in the world may be fuzzy, inconsistent or uncertain to us — to observers or even to scientific theorists.

Yet (to take just a single example from the world of philosophical logic) the philosopher and logician Bryson Brown helpfully states the…


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This is a response to Sabine Hossenfelder’s article, ‘Electrons Don’t Think’.

Sabine Hossenfelder (that link is to her Medium account) is a physicist and a Research Fellow at the Frankfurt Institute for Advanced Studies. She’s also a popular writer on science and a presenter.

As for her article itself: it’s crude and rhetorical. My piece is also (somewhat) rhetorical. It is so because I believe that my own unacademic phrases are a fitting response to Hossenfelder’s own.

But first things first.

I’m not a panpsychist.

I’m not even particularly sympathetic to panpsychism and find some panpsychists’ ideas absurd. Nonetheless, none of this warrants Hossenfelder’s philosophically ignorant and sarcastic article on panpsychism. …


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… or do animals deploy concepts?

There’s been a long-running and controversial debate as to whether or not animals are conscious (or have experiences). I believe that it’s most certainly the case that animals are conscious — or at least the “higher animals” are. Of course that claim can’t be proved. That said, I can’t prove that my friends are conscious. So there are no proofs — or even conclusive demonstrations — when it comes to whether any given biological animal — including a human being — is conscious or not.

There’s also the debate as to whether or not animals have beliefs, are intelligent, are capable of thought, etc. This often — mainly — depends on definitions. That is, if the word “belief” (or “thought”) is defined in one way, then animals will be deemed to have beliefs (or to think). If, on the other hand, such a word is defined in another way, then animals won’t be deemed to have beliefs (or to think). …


Essays on Logic

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I’ve been informed that there’s a lack of parenthesis for the second use of symbol F in Leibniz’s law above. I’ve corrected it in the text itself.

Firstly we can say that logical identity is reflexive. In other words, everything (or every thing) is identical to itself. In symbols:

x (x = x)

To translate:

For every x (or for every thing), x must equal x (or everything must be identical to itself). That is, everything has the (quasi)-relation of self-identity..

The logic of identity is also symmetrical. In symbols:

If a = b, then b = a

This can also be expressed in quantificational logic:

x y ((x = y) ⊃ (y = x))

To translate:

For every x, and for every y, if x equals — or is identical to — y, then it must also be the case that y is identical to x. …


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I always had a problem with the term “meaningless” as it was used by the logical positivists in the 1920s, 1930s, and 1940s. My problem existed even though I sympathised with (some of) the spirit of logical positivism. (I still do.) It seemed to me that classing statements as “meaningless” is problematic and somewhat pompous. And even when I came to realise that the word “meaningless” had a highly-technical meaning, I still found it suspect.

Still, once the details are out of the way, it can be seen that the use of word “meaningless” is not as problematic as it initially sounds. …


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On February the 12th, 2002, the then Secretary of Defense of the United States, Donald Rumsfeld, stated the following words (as captured in a YouTube video here):

“[A]s we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns — the ones we don’t know we don’t know.”

At first I wasn’t going to tackle this passage because some people may believe that I have some political sympathy for Donald Rumsfeld and what he said. However, since this passage was spoken some 18 years ago, and was spoken by someone who no longer has a prominent position in politics, I can’t see why that should be the case. Besides which, I shan't refer to the political context of Rumsfeld’s words at all (although I will offer a little background). …


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Left: E. Brian Davies (Photo: Wikimedia Commons). Right: Front Cover of Davies’ Book “Science in the Looking Glass”

Edward Brian Davies was born in 1944. He is a Fellow of the Royal Society and was Professor of Mathematics at King’s College London (1981–2010). Davies has written papers on spectral theory, non-self-adjoint operators, operator theory/functional analysis, elliptic partial differential operators, Schrodinger operators (in quantum theory) and so on. He was awarded a Gauss Lectureship by the German Mathematical Society in 2010.

This commentary is on the relevant parts of Davies’s book, Science in the Looking Glass. The following is not a book review.

A Short Introduction to Mathematical Empiricism

E. Brian Davies’s own position on mathematics (or at least on numbers) is generally called mathematical empiricism.


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Left: Badiou’s book ‘In Praise of Mathematics’. Right: Alain Badiou in 2011 (Photo: Wikimedia Commons)

Alain Badiou (1937-) is a French philosopher. At one point he was the chair of Philosophy at the École normale supérieure (ENS) and founder (with Michel Foucault, Gilles Deleuze and Jean-François Lyotard) of the Faculty of Philosophy at the Université de Paris VIII. He’s now René Descartes Chair and Professor of Philosophy at The European Graduate School. Badiou has also been involved in a politics and political organisations since early in his life. Indeed he has often commented on both French and global political affairs.

More relevantly to this piece, Badiou has a strong mathematical background. He’s the son of the mathematician Raymond Badiou (1905–1996). And, according to Badiou himself, by 1967 he “already had a solid grounding in mathematics and logic”. Badiou again describes his own history when he tells us that he studied “contemporary mathematics in greater depths by taking the first two years of university math”. He then goes on to say that “[t]his was from 1956 to 1958, my first two years at at the École Normale Supérieure”. …

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