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19335
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Reasonings have a natural ordering in God's understanding, but only a temporal order in ours [Leibniz]
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Full Idea:
All reasonings are eminent in God, and they preserve an order among themselves in his understanding as well as in ours; but for him this is just an order and a priority of nature, whereas for us there is a priority of time.
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From:
Gottfried Leibniz (The Theodicy [1710], p.192), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.III
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A reaction:
This view is found in Frege, and seems to be the hallmark of rationalist philosophy. There is an apriori assumption that reality has a rational order, so that pure reason is a tool for grasping it. Lewis's 'mosaic' of experiences has no order.
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8729
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Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro]
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Full Idea:
Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements.
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From:
Stewart Shapiro (Thinking About Mathematics [2000], 1.2)
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A reaction:
There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle.
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8763
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The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
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Full Idea:
It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures.
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From:
Stewart Shapiro (Thinking About Mathematics [2000], 10.2)
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A reaction:
The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed.
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8762
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Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
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Full Idea:
Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3.
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From:
Stewart Shapiro (Thinking About Mathematics [2000], 10.2)
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A reaction:
See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them.
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8749
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Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
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Full Idea:
Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names?
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From:
Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1)
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A reaction:
Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up.
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8750
|
Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
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Full Idea:
Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences.
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From:
Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2)
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A reaction:
This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare.
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8753
|
Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
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Full Idea:
Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle.
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From:
Stewart Shapiro (Thinking About Mathematics [2000], 7.1)
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A reaction:
The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them.
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8731
|
Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]
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Full Idea:
I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists.
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From:
Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1)
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A reaction:
In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football.
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19367
|
Saying we must will whatever we decide to will leads to an infinite regress [Leibniz]
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Full Idea:
As for volition itself, to say that it is the object of free will is incorrect. We will to act, strictly speaking, and we do not will to will, else we should still say we will to have the will to will, and that would go on to infinity.
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From:
Gottfried Leibniz (The Theodicy [1710], p.151), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 4.IV
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A reaction:
This strikes me as an elementary difficulty which most fans of free will appear to evade. Thoughts just arise in us, and some of them are volitions. We can say there is then a 'gap' (Searle) where we choose, but what happens in the gap?
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19351
|
Perfections of soul subordinate the body, but imperfections of soul submit to the body [Leibniz]
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Full Idea:
Insofar as the soul has perfection ...God has accommodated the body to the soul, and has arranged beforehand that the body is impelled to execute its orders. Insofar as it is imperfect and confused, God accommodates soul to body, swayed by passions.
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From:
Gottfried Leibniz (The Theodicy [1710], p.159), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 3.IV
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A reaction:
Perkins says this is the nearest Leibniz gets to the idea of interaction between body and soul. Perfection and confusion are on a continuum for Leibniz. With such speculations I always wonder how these things can be known. How perfect is my mind?
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19346
|
Most people facing death would happily re-live a similar life, with just a bit of variety [Leibniz]
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Full Idea:
I believe there would be few persons who, being at the point of death, were not content to take up life again, on condition of passing through the same amount of good and evil, provided that it were not the same kind.
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From:
Gottfried Leibniz (The Theodicy [1710], p.130), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.IV
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A reaction:
Nice challenge. People who refuse the offer are not necessarily suicidal. He's probably right, but Leibniz doesn't recognise the factor of boredom. Look up the suicide note of the actor George Sanders! One life may be enough.
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19340
|
Metaphysical evil is imperfection; physical evil is suffering; moral evil is sin [Leibniz]
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Full Idea:
Evil may be taken metaphysically, physically, and morally. Metaphysical evil consists in mere imperfection, physical evil is suffering, and moral evil is sin.
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From:
Gottfried Leibniz (The Theodicy [1710], p.136), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.IV
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A reaction:
There seem to be plenty of imperfections in the world which don't look like evil. Or do you only declare it to be an imperfection because it seems to be evil (by some other standard)? Human evil comes from ignorance, so metaphysical explains moral.
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5121
|
Basing ethics on flourishing makes it consequentialist, as actions are judged by contributing to it [Harman]
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Full Idea:
Basing ethics on human flourishing tends towards utilitarianism or consequentialism; actions, character traits, laws, and so on are to be assessed with reference to their contributions to human flourishing.
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From:
Gilbert Harman (Human Flourishing, Ethics and Liberty [1983], 9.2.2)
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A reaction:
This raises the question of whether only virtue can contribute to flourishing, or whether a bit of vice might be helpful. This problem presumably pushed the Stoics to say that virtue itself is the good, rather than the resulting flourishing.
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19327
|
The intelligent cause must be unique and all-perfect, to handle all the interconnected possibilities [Leibniz]
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Full Idea:
The intelligent cause ought to be infinite in all ways, and absolutely perfect in power, in wisdom, and in goodness, since it relates to all that which is possible. Also, since all is connected together, there is no ground for admitting more than one.
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From:
Gottfried Leibniz (The Theodicy [1710], p.128), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.II
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A reaction:
Notice that Leibniz's possible worlds seem to be all connected together, unlike David Lewis's worlds, which are discrete. Personally I suspect that all perfections will lead to contradiction, though Leibniz strongly argues against it.
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19330
|
If justice is arbitrary, or fixed but not observed, or not human justice, this undermines God [Leibniz]
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Full Idea:
The three dogmas (1) that the nature of justice is arbitrary, (2) it is fixed, but not certain God will observe it, or (3) the justice we know is not that which God observes, destroy our confidence in the love of God.
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From:
Gottfried Leibniz (The Theodicy [1710], p.237), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.III
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A reaction:
Leibniz proceeds to carefully refute these three responses to the dilemma about how justice relates to God.
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19329
|
The laws of physics are wonderful evidence of an intelligent and free being [Leibniz]
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|
Full Idea:
These admirable laws [of physics] are wonderful evidence of an intelligent and free being, as opposed to the system of absolute and brute necessity, advocated by Strato and Spinoza.
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From:
Gottfried Leibniz (The Theodicy [1710], p.332), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.II
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A reaction:
Note the swipe at Spinoza. Leibniz defends the absolute necessities residing in God, but is too polite to call those 'brute', though personally I can't see the difference. But he says the laws arise from 'perfection and order', not from God's necessity.
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19337
|
How can an all-good, wise and powerful being allow evil, sin and apparent injustice? [Leibniz]
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Full Idea:
There is this question of natural theology, how a sole Principle, all-good, all-wise and all-powerful, has been able to admit evil, and especially to permit sin, and how it could resolve to make the wicked often happy and the good unhappy?
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From:
Gottfried Leibniz (The Theodicy [1710], p.098), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.IV
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A reaction:
His answer is, roughly, that there is an unavoidable trade-off, which humans cannot fully understand. Personally I would say that if there is a God, the evidence for his benevolence towards humanity is not encouraging.
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19345
|
Being confident of God's goodness, we disregard the apparent local evils in the visible world [Leibniz]
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Full Idea:
Being made confident by demonstrations of the goodness and the justice of God, we disregard the appearances of harshness and justice which we see in this small portion of his Kingdom that is exposed to our gaze.
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From:
Gottfried Leibniz (The Theodicy [1710], p.120), quoted by Franklin Perkins - Leibniz: Guide for the Perplexed 2.IV
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A reaction:
Hm. If this locality is full of evils, and the rest of it is much better, how come we are stuck in this miserable corner of things? God is obliged to compromise, but did he select us to get the worst of it?
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