|
7485
|
For Pythagoreans 'one' is not a number, but the foundation of numbers [Pythagoras, by Watson]
|
|
Full Idea:
For Pythagoreans, one, 1, is not a true number but the 'essence' of number, out of which the number system emerges.
|
|
From:
report of Pythagoras (reports [c.530 BCE], Ch.8) by Peter Watson - Ideas Ch.8
|
|
A reaction:
I think this is right! Counting and numbers only arise once the concept of individuality and identity have arisen. Counting to one is no more than observing the law of identity. 'Two' is the big adventure.
|
|
10529
|
If Hume's Principle can define numbers, we needn't worry about its truth [Fine,K]
|
|
Full Idea:
Neo-Fregeans have thought that Hume's Principle, and the like, might be definitive of number and therefore not subject to the usual epistemological worries over its truth.
|
|
From:
Kit Fine (Precis of 'Limits of Abstraction' [2005], p.310)
|
|
A reaction:
This seems to be the underlying dream of logicism - that arithmetic is actually brought into existence by definitions, rather than by truths derived from elsewhere. But we must be able to count physical objects, as well as just counting numbers.
|
|
10530
|
Hume's Principle is either adequate for number but fails to define properly, or vice versa [Fine,K]
|
|
Full Idea:
The fundamental difficulty facing the neo-Fregean is to either adopt the predicative reading of Hume's Principle, defining numbers, but inadequate, or the impredicative reading, which is adequate, but not really a definition.
|
|
From:
Kit Fine (Precis of 'Limits of Abstraction' [2005], p.312)
|
|
A reaction:
I'm not sure I understand this, but the general drift is the difficulty of building a system which has been brought into existence just by definition.
|
|
22110
|
Demonstration provides depth of understanding and explanation (rather than foundations) [Kretzmann/Stump]
|
|
Full Idea:
According to Aquinas, what demonstration provides is not so much knowledge as conceived by foundationalists as depth of understanding and explanatory insight.
|
|
From:
Kretzmann/Stump (Aquinas, Thomas [2005]), quoted by Kretzmann/Stump - Aquinas, Thomas 11
|
|
A reaction:
It was noticeable that Aristotle didn't make clear what demonstration aims to achieve, and he didn't employ it elsewhere in his writings. We aim for understanding, not for well grounded propositions. Understanding needs implications and mechanisms.
|
|
10527
|
An abstraction principle should not 'inflate', producing more abstractions than objects [Fine,K]
|
|
Full Idea:
If an abstraction principle is going to be acceptable, then it should not 'inflate', i.e. it should not result in there being more abstracts than there are objects. By this mark Hume's Principle will be acceptable, but Frege's Law V will not.
|
|
From:
Kit Fine (Precis of 'Limits of Abstraction' [2005], p.307)
|
|
A reaction:
I take this to be motivated by my own intuition that abstract concepts had better be rooted in the world, or they are not worth the paper they are written on. The underlying idea this sort of abstraction is that it is 'shared' between objects.
|
|
3053
|
Pythagoras taught that virtue is harmony, and health, and universal good, and God [Pythagoras, by Diog. Laertius]
|
|
Full Idea:
Pythagoras taught that virtue is harmony, and health, and universal good, and God.
|
|
From:
report of Pythagoras (reports [c.530 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 08.1.19
|
|
A reaction:
I like the link with health, because I consider that a bridge over the supposed fact-value gap. Very Pythagorean to think that virtue is harmony. Plato liked that thought.
|
|
5244
|
For Pythagoreans, justice is simply treating all people the same [Pythagoras, by Aristotle]
|
|
Full Idea:
Some even think that what is just is simple reciprocity, as the Pythagoreans maintained, because they defined justice simply as having done to one what one has done to another.
|
|
From:
report of Pythagoras (reports [c.530 BCE], 28) by Aristotle - Nicomachean Ethics 1132b22
|
|
A reaction:
One wonders what Pythagoreans made of slavery. Aristotle argues that officials, for example, have superior rights. The Pythagorean idea makes fairness the central aspect of justice, and that must at least be partly right.
|