28

After a bit of searching, I found some promising leads (and quite a few consistent descriptions) which suggest that Russell thought Gödel's results were of cardinal importance, but misunderstood their implications. In particular, he thought that Gödel's result essentially entailed that Peano Arithmetic was inconsistent rather than incomplete; but ...


26

'The ancient Greek philosophers' is rather a wide reference class. We don't really know enough about the historical Socrates to be sure about his views on the traditional gods. Plato In the Republic Plato removes from his educational curriculum all unseemly stories about the gods but my impression is that he does this, not because he believes in the gods ...


20

Here is an example of philosophy helping a breakthrough in mathematics (in differential topology). The breakthrough happened last year, the philosophy that helped it come into existence happened 200 years ago, via a formalization suggested in the last two decades. A long-standing open problem in differential topology and in mathematical physics was the ...


20

I will enlarge upon Nuclear Wang's reply by turning the question around a bit, as follows. When did natural philosophers (also known as scientists) decide that philosophy was not useful, and why? I believe this change in thinking about the natural world occurred along with the development of the scientific method, when its practitioners grasped the idea that ...


19

I'd go with Simon Blackburn's Think: A Compelling Introduction to Philosophy (Cambridge University Press 1999). Here's the blurb: This is a book about the big questions in life: knowledge, consciousness, fate, God, truth, goodness, justice. It is for anyone who believes there are big questions out there, but does not know how to approach them. Think ...


19

Here is Cantor in his own words (from his influential 1887 letter to Weierstrass): "I begin from the supposition of a linear magnitude ζ which is so small that its product by n , ζ · n, for every finite whole number n however great is smaller than unity, and then prove, from the concept of a linear magnitude and with the help of certain propositions from ...


19

It exists. It is called meta-philosophy. It includes topics such as the definition and scope of philosophy, the methods of philosophy, and the role of philosophers.


18

I am of Indian ancestry but 3rd generation in America. I have been exposed to both East and Western philosophy/science in my family life but exposed to Western science due to my graduate school training. I identify far more with the Western approach culturally but I hope I can provide some insight. Cultural, academic, and scientific standards have shaped ...


17

There is a non-philosophical component to this which I think the other answers touched on, but did not quite get to. The years leading up to Socrates' trial were hard on Athens. Athens had just lost the Peloponnesian War against its long-time rival Sparta, and Sparta had installed the Thirty Tyrants. Though their rule only lasted 13 months, it was extremely ...


17

Because there was a calculus for one-place predicates, Aristotle's syllogistic, roughly equivalent to monadic predicate calculus. Aristotle does discuss "relatives" in Categories, which refer to multi-place relations, or rather to objects entering them. What will later be called oblique syllogisms involving relatives is mentioned in passing in ...


16

I'd say it started around the 1950 and got off the ground around 1980 ;) That is to say, the divide as been introduced as a fighting word from the beginning; it is more about asserting the divide than about giving an adequate picture of the philosophical landscape. And it is really not advisable to do proper history of philosophy by using fighting words! ...


16

Is it that New Atheism isn't philosophy? Yes -- it is not a philosophy; wikipedia describes it as "a social and political movement" first associated with Sam Harris, an American writer who holds a B.A. in philosophy and a Ph.D in neuroscience but does not practice either professionally in the sense of having academic tenure. Harris's works appear to be ...


16

Nietzsche mocked German idealists at length, but I think calling him a materialist is a bridge too far, same as for all his anti-Christianity it is not clear that he was an atheist. He inherited his metaphysics from Schopenhauer, transforming his World Will into will to power, who can be seen as irrationalizing Hegel's Absolute Geist with a side of that "...


15

Peter Kreeft has written a book on this very topic. Would He give a lecture at Harvard, or engage in a long Socratic dialog in Plato's Academy, or write a critique of Kant's Critique of Pure Reason? Obviously not. And everyone knows that. That is "trivially true." In another sense, Jesus was a philosopher, but this second sense is also trivial. ...


15

Philosophy became more limited in scope during the 19th century. In the time of the ancient Greeks, philosophy encompassed all forms of knowledge, but specialized subfields of physics, medicine, astronomy, and so on later emerged and became distinct from philosophy. In the past, philosophy encompassed pretty much all knowledge, but today, it's a much more ...


15

In the old joke, a college president complains to his physics professors that they are always asking for expensive equipment. All the mathematics teachers need is a wastebasket, he says, and the philosophers don't even need that. There has always been a feeling among many that philosophy is useless or worse, and one can find this sentiment expressed not only ...


14

Let me give you a historical background first. Until the end of 19th century logic was almost exclusively associated with Aristotelian logic, the syllogistic. This logic did not have quantifiers, or even propositional variables, in other words it was too weak to support even arithmetic, let alone the rest of mathematics (Chrysippus, an ancient Stoic, and ...


14

You can see: Irving Anellis, The Genesis of the Truth-Table Device (2004) as well as: Irving Anellis, Peirce's Truth-functional Analysis and the Origin of the Truth Table (2012). Before Bertrand Russell (Harvard logic course: 1914) and Ludwig Wittgenstein (Russell and Wittgenstein's manuscript dated 1912; see also: Tractatus (1921), 4.31 and 4.442 for ...


14

Logical positivism does not deny human emotion. It simply reassigns its role. Ethical, aesthetic or religious judgements, for example, fulfil the role of expressing or eliciting emotion - and not, emphatically not, of truth-bearing. Since ethical, aesthetic and religious judgements definitely occur and since, equally definitely for logical positivists, they ...


13

Yes, philosophers have made some recent contributions to other fields. And just countering gross generalization with gross generalization, I believe it is generally far from true, as Aaronson suggests above, that philosophers working right now on applications to other fields are ignorant of those other fields. A few preliminaries: I'll answer about the ...


13

It would help to divide up philosophy into some component parts. Logic has certainly made a great deal of progress. Frege's logic is a huge step forward from everything that had gone before and provided the impetus for further advances that have turned the subject into something that Aristotle or Mill would scarecely recognise. We now understand concepts ...


11

Monism does not necessarily posit that there is one thing that is the whole so much as everything is really one kind of thing. Just to give one quick example to show it's less crazy than you think in modern thought. Many of todays leading scientists and quite a large percent of philosophers are materialists. Materialists are a type of monist as they ...


11

Popper objected to Marxism because Marxism is historicist: Marxism claims that it can predict the future course of history. When I discuss Marxism below, I'm discussing what Popper said about Marxism without necessarily claiming Popper was right. (I think he was far too generous to Marx in many respects. In particular, both Popper and Marx were very badly ...


11

The concept of infinitesimal small and infinitely large numbers has been been formalized by the mathematical domain of non-standard analysis. The field of rationals (QQ,+,*) embedds into the ring (Omega_QQ,+,*). Elements of the latter are the equivalence classes of sequences of rational numbers; two sequences are considered equivalent when their ...


11

To answer this, its helpful to have some background on Whitehead's understanding of rationalism. In Process and Reality (42) Whitehead defines rationalism as the hope that we can create a general theory, such that everything we find in experience can be held up as an example of that theory. He calls it a "hope" since Whitehead argues there will ...


11

Whether Plato's character of Socrates or Plato himself believed in a God or many Gods is not perfectly clear. Additionally, we can't ascribe any sort of belief to the historical Socrates; we just don't know enough about his life that doesn't come from Plato or Xenophon. For the rest of this answer I'll say "Socrates" instead of "Plato's ...


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