Combining Philosophers

All the ideas for Cardinal/Hayward/Jones, Hippocrates and R Kaplan / E Kaplan

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13 ideas

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
Using Choice, you can cut up a small ball and make an enormous one from the pieces [Kaplan/Kaplan]
     Full Idea: The problem with the Axiom of Choice is that it allows an initiate (by an ingenious train of reasoning) to cut a golf ball into a finite number of pieces and put them together again to make a globe as big as the sun.
     From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 9)
     A reaction: I'm not sure how this works (and I think it was proposed by the young Tarski), but it sounds like a real problem to me, for all the modern assumptions that Choice is fine.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
1 and 0, then add for naturals, subtract for negatives, divide for rationals, take roots for irrationals [Kaplan/Kaplan]
     Full Idea: You have 1 and 0, something and nothing. Adding gives us the naturals. Subtracting brings the negatives into light; dividing, the rationals; only with a new operation, taking of roots, do the irrationals show themselves.
     From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 1 'Mind')
     A reaction: The suggestion is constructivist, I suppose - that it is only operations that produce numbers. They go on to show that complex numbers don't quite fit the pattern.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
The rationals are everywhere - the irrationals are everywhere else [Kaplan/Kaplan]
     Full Idea: The rationals are everywhere - the irrationals are everywhere else.
     From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 1 'Nameless')
     A reaction: Nice. That is, the rationals may be dense (you can always find another one in any gap), but the irrationals are continuous (no gaps).
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
'Commutative' laws say order makes no difference; 'associative' laws say groupings make no difference [Kaplan/Kaplan]
     Full Idea: The 'commutative' laws say the order in which you add or multiply two numbers makes no difference; ...the 'associative' laws declare that regrouping couldn't change a sum or product (e.g. a+(b+c)=(a+b)+c ).
     From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 2 'Tablets')
     A reaction: This seem utterly self-evident, but in more complex systems they can break down, so it is worth being conscious of them.
'Distributive' laws say if you add then multiply, or multiply then add, you get the same result [Kaplan/Kaplan]
     Full Idea: The 'distributive' law says you will get the same result if you first add two numbers, and then multiply them by a third, or first multiply each by the third and then add the results (i.e. a · (b+c) = a · b + a · c ).
     From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 2 'Tablets')
     A reaction: Obviously this will depend on getting the brackets right, to ensure you are indeed doing the same operations both ways.
11. Knowledge Aims / C. Knowing Reality / 2. Phenomenalism
The phenomenalist says that to be is to be perceivable [Cardinal/Hayward/Jones]
     Full Idea: Where the idealist says that to be (i.e. to exist) is to be perceived, the phenomenalist says that to be is to be perceivable.
     From: Cardinal/Hayward/Jones (Epistemology [2004], Ch.4)
     A reaction: This is a nice phenomenalist slogan to add to Mill's well known one (Idea 3583). Expressed in this form, it looks false to me. What about neutrinoes? They weren't at all perceivable until recently. Maybe some physical stuff can never be perceived.
Linguistic phenomenalism says we can eliminate talk of physical objects [Cardinal/Hayward/Jones]
     Full Idea: Linguistic phenomenalism argues that it is possible to remove all talk of physical objects from our speech with no loss of meaning.
     From: Cardinal/Hayward/Jones (Epistemology [2004], Ch.4)
     A reaction: I find this proposal unappealing. My basic objection is that I cannot understand why anyone would refuse to even contemplate the question of WHY I am having a given group of consistent experiences, of (say) a table kind.
If we lack enough sense-data, are we to say that parts of reality are 'indeterminate'? [Cardinal/Hayward/Jones]
     Full Idea: The problem with taking sense-data as basic is that some data can appear indeterminate. If we can't discern the colour of someone's eyes, or the number of sides of a complex figure, are we to say that there is no fact about those things?
     From: Cardinal/Hayward/Jones (Epistemology [2004], Ch.4)
     A reaction: I like that. How many electrons are there in the sun? Such things cannot just be reduced to talk of sense-data, as there is obviously a vast gap between the data and the facts. As usual, ontology and epistemology must be kept separate.
12. Knowledge Sources / B. Perception / 2. Qualities in Perception / c. Primary qualities
Primary qualities can be described mathematically, unlike secondary qualities [Cardinal/Hayward/Jones]
     Full Idea: All the primary qualities lend themselves readily to mathematical or geometric description. ...but it seems that secondary qualities are less amenable to being represented mathematically.
     From: Cardinal/Hayward/Jones (Epistemology [2004], Ch.4)
     A reaction: As a believer in the primary/secondary distinction, I welcome this point. This is either evidence for the external reality of primary qualities, or an interesting observation about maths. Do we make the primary/secondary distinction because we do maths?
An object cannot remain an object without its primary qualities [Cardinal/Hayward/Jones]
     Full Idea: An object cannot lack shape, size, position or motion and remain an object.
     From: Cardinal/Hayward/Jones (Epistemology [2004], Ch.4)
     A reaction: This points towards the essentialist view (see Idea 5453). This does raise the question of whether an object could lose its colour with impugnity, or the quality of sound that it makes when struck.
13. Knowledge Criteria / B. Internal Justification / 5. Coherentism / c. Coherentism critique
My justifications might be very coherent, but totally unconnected to the world [Cardinal/Hayward/Jones]
     Full Idea: My beliefs could be well justified in coherentist terms, while not accurately representing the world, and my system of beliefs could be completely free-floating.
     From: Cardinal/Hayward/Jones (Epistemology [2004], Ch.3)
     A reaction: This nicely encapsulates to correspondence objection to coherence theory. One thing missing from the coherence account is that beliefs aren't chosen for their coherence, but are mostly unthinkingly triggered by experiences.
14. Science / C. Induction / 3. Limits of Induction
The first million numbers confirm that no number is greater than a million [Kaplan/Kaplan]
     Full Idea: The claim that no number is greater than a million is confirmed by the first million test cases.
     From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 2 'Intro')
     A reaction: Extrapolate from this, and you can have as large a number of cases as you could possibly think of failing to do the inductive job. Love it! Induction isn't about accumulations of cases. It is about explanation, which is about essence. Yes!
17. Mind and Body / E. Mind as Physical / 1. Physical Mind
All of our happiness and misery arises entirely from the brain [Hippocrates]
     Full Idea: Men ought to know that from the brain, and from the brain alone, arise our pleasures, joys, laughter and jests, as well as our sorrow, pains, griefs and tears.
     From: Hippocrates (Hippocrates of Cos on the mind [c.430 BCE], p.32)
     A reaction: If this could be assertedly so confidently at that date, why was the fact so slow to catch on? Brain injuries should have convinced everyone.