Combining Texts

All the ideas for 'Mathematics without Numbers', 'Lectures on Ethics' and 'Sententia on 'Posterior Analytics''

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5 ideas

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism

8698	Modal structuralism says mathematics studies possible structures, which may or may not be actualised [Hellman, by Friend]
	Full Idea: The modal structuralist thinks of mathematical structures as possibilities. The application of mathematics is just the realisation that a possible structure is actualised. As structures are possibilities, realist ontological problems are avoided.
	From: report of Geoffrey Hellman (Mathematics without Numbers [1989]) by Michèle Friend - Introducing the Philosophy of Mathematics 4.3
	A reaction: Friend criticises this and rejects it, but it is appealing. Mathematics should aim to be applicable to any possible world, and not just the actual one. However, does the actual world 'actualise a mathematical structure'?

9557	Statements of pure mathematics are elliptical for a sort of modal conditional [Hellman, by Chihara]
	Full Idea: Hellman represents statements of pure mathematics as elliptical for modal conditionals of a certain sort.
	From: report of Geoffrey Hellman (Mathematics without Numbers [1989]) by Charles Chihara - A Structural Account of Mathematics 5.3
	A reaction: It's a pity there is such difficulty in understanding conditionals (see Graham Priest on the subject). I intuit a grain of truth in this, though I take maths to reflect the structure of the actual world (with possibilities being part of that world).

10263	Modal structuralism can only judge possibility by 'possible' models [Shapiro on Hellman]
	Full Idea: The usual way to show that a sentence is possible is to show that it has a model, but for Hellman presumably a sentence is possible if it might have a model (or if, possibly, it has a model). It is not clear what this move brings us.
	From: comment on Geoffrey Hellman (Mathematics without Numbers [1989]) by Stewart Shapiro - Philosophy of Mathematics 7.3
	A reaction: I can't assess this, but presumably the possibility of the model must be demonstrated in some way. Aren't all models merely possible, because they are based on axioms, which seem to be no more than possibilities?

13. Knowledge Criteria / B. Internal Justification / 5. Coherentism / a. Coherence as justification

22109	The fullest knowledge places a conclusion within an accurate theory [Aquinas, by Kretzmann/Stump]
	Full Idea: Having 'scientia' is the fullest possible human cognition, by which one situates the fact expressed by a conclusion in an explanatory theory that accurately maps metaphysical or physical reality.
	From: report of Thomas Aquinas (Sententia on 'Posterior Analytics' [1269], 1.2.9, 1.5.7) by Kretzmann/Stump - Aquinas, Thomas 11
	A reaction: That is a perfect statement of my concept of knowledge. Explanatory theories must specify the essential natures of the entities involved. We don't aim for 'knowledge', we aim for the 'fullest possible cognition'. This account extend's Aristotle's.

25. Social Practice / F. Life Issues / 4. Suicide

19739	The maxim for suicide is committed to the value of life, and is thus contradictory [Kant]
	Full Idea: If my maxim is to shorten my life if its continuance threatens more evil than pleasure ...it is seen that a system of nature by whose law the feeling intended to further life should actually destroy life would contradict itself, and could not subsist.
	From: Immanuel Kant (Lectures on Ethics [1780], 422:53)
	A reaction: [compressed] I take it this means that a potential suicide is assessing what is best for life, and is therefore implicitly committed to life. Not persuasive! Should we not terminate the life of a mass murderer in mid-crime?