I'm not sure what your tutor was getting at either. If your tutor meant that one can always enlarge a set of premises to make it an inconsistent set, that's obviously true: simply add the negation of one of the premises already in the set. If he meant that any axiom system with infinitely many premises (say, one that employs an axiom schema) is inconsistent, then there's no reason to believe that.

There is nothing stopping you from defining an analytic theorem of a formal system to be one whose derivation requires appeal to at least one member of a designated subset of axioms. But on what basis are you deciding to single out that particular subset of axioms? If you say you're being guided by the fact that those particular axioms express truths about meanings, whereas other axioms express substantive truths about the world, then you owe an explanation of what that distinction amounts to -- and arguably, that will be no easier to give than an outright analysis of "analytic". (You might also look at W.V. Quine's discussion of Semantic Postulates in his paper "Two Dogmas of Empiricism.")

Thank you for your question. For a long while in the history of philosophy it was thought that what was conceivable was a good indication of what was possible. Descartes is a good example of this way of thinking, though he was careful to require that not any old conceiving of a thing showed it to be possible. Rather he required that the conceiving had to be "clear and distinct", meaning roughly that it had to pass the most stringent standards we can muster to make sure the conceiving is coherent (i.e., not subtly self-contradictory). In the middle of the 20th century this methodology began to break down. For instance, in the Sixties Hilary Putnam distinguished between concepts and properties, making clear that our concepts of things like gold may not reveal its true properties. Similarly, Kripke's notion a decade later of "natural kinds" made room for the possibility that what is "metaphysically possible" may not correspond to that is conceivable.

This issue is still a topic of intense debate. Some philosophers in the last decade or so have argued that conceivability considerations have *some* force in determining what is genuinely possible. For instance, see Frank Jackson's, From Metaphysics to Ethics: A Defense of Conceptual Analysis, Oxford University Press, 1997. I should mention that conceivability does seem an important tool for many fields, not just philosophy: for instance, physics uses thought experiments regularly. Those "experiments" are constrained by what we know of the laws of physics, but philosophers' thought experiments can take into account all we know also about the empirical world. Imagine, further, how hard it would be to reason in ethics or political philosophy without being about to construct thought experiments! The moral here is that conceivability considerations are not aimed at finding out about our minds, but rather are our attempts to use common sense, albeit fallibly, to find out about the world.

Mitch Green

"Shoulds" in philosophy are a tough sell. And in particular, the idea that philosophers "should not" try to overturn ordinary notions is one that's regularly challenged by philosophers. For example: many philosophers have argued that there is no such thing as a "self." Some philosophers have argued that the ordinary notion of belief is incoherent. And challenges to the idea of time, to take your example, have come from within philosophy itself; McTaggart's famous article "The Unreality of Time" offered purely philosophical arguments for abandoning our familiar ways of thinking about time.

It would be not just hard but perverse to argue that philosophy should never challenge our ordinary conceptions -- even if the challenge runs very deep. After all, sometimes we are confused, and even when we're not, there's often something to be learned from meeting the challenge. That said, some attacks on ordianry notions may take those notions to carry more baggage than they really do. The case of the "self" may be a good example. Ordinary people seldom use phrases like "the self," though they do use words like "myself" and "yourself," and while people may say peculiar things about "the self" when pressed by philosophers, it's not clear that the extra stuff plays much of a role in their day-to-day thinking.

Poetry can certainly be used to express profound ideas and attitudes concerning (for want of a better expression) 'the human condition'. These ideas can affect the reader's soul in a powerful way, helped along by the captivating power of the medium itself. And examples of poetry that might be regarded as 'philosophical' in this sense are innumerable. Indeed, one might make a case for claiming that it's the norm rather than the exception, and that this is the primary aspiration of most of the greatest poetry in history, from Homer to Dante to Sylvia Plath.

But does this really count as philosophy? For some people, this is precisely what the best and most important kind of philosophy consists in. For others, however, and particularly within English-speaking academia, philosophy is more a matter of highly technical and abstract theories about the structure of reality, the nature of cognition, and things of that sort. And yet, as it turns out, those kinds of theories have been explored in verse form too. One might compare this with the way in which philosophers from Plato to Berkeley to David Lewis have opted to present their ideas and arguments in the form of witty dialogues. Either approach brings, among other potential advantages, that of simply engaging the reader more effectively than yet another dry prose treatise might.

For an example, consider Lucretius's De Rerum Natura, an epic poem in six books from the first century BC. Lucretius set out and argued for, among other things: an atomistic physics, a plurality of worlds and extraterrestrial life, a theory of natural selection, a materialist account of the soul, a vigorous critique of religion, an account of the origins of human society out of a state of nature, a study of meteorology, and, believe it or not, a discussion of sexual positions. And all in Latin verse. Admittedly, that was then and this is now. Professional philosophers and scientists tend not to express themselves in verse any more -- I can't think of any recent examples (though perhaps others might). But, even into the early modern period, they were still doing so. See, for instance, the book-length philosophical poems, Nosce Teipsum by Sir John Davies (1599) or Psychodia Platonica by Henry More (1642). Some of the scansion and rhyme might have been a bit dodgy, but these authors did nevertheless feel that verse was an appropriate medium for the expression of serious metaphysical and epistemological theories and arguments. There's certainly no incompatibility between the poetic medium and even the most technical kind of philosophy.

The fact that we take pleasure in performing good acts and even perhaps expect to take pleasure in good acts doesn't mean that we perform them in order to produce this pleasure for ourselves. To assume that we must be acting selfishly in such cases would be to confuse the consequences we intend and those we merely foresee. We foresee that doing good deeds will have pleasure as a by-product but this is not why most of us perform them.

Indeed, this is no accident. For we can ask why we take pleasure in performing good deeds, and the answer is presumably because we enjoy doing what we believe to be good or right. The pleasure is consequential on the perception of doing what's good or right. If we didn't already have a desire to be good, we wouldn't take pleasure in doing what we regard as good. But that means that the desire to do good is prior to the pleasure, not the other way around. But then the pleasure is a by-product of the desire to be good, which is doing the real explanatory work. (Similar remarks apply to those who are pained at doing bad deeds. The pain is consequential on the perception of doing wrong and the prior desire to avoid wrongdoing.)

Some people may behave morally for purely prudential reasons, as a way of avoiding legal or social sanctions. Their motivation may be selfish or otherwise morally suspect. But taking pleasure in good deeds does not show that your real motivation is selfish or taint the moral quality of your acts.

An interesting question. The word comes from 'elite', obviously, and ultimately from Latin by way of French; originally it meant the 'chosen' or 'elected'. So, in a democracy (and for the purposes of this answer I'll assume that's the position we are concerned with) our leaders are indeed the 'elite', and insofar as we think there should be elections and that the winner of the election should be the new leader, all voters are 'elitist'!

But that is disingenuous, because that original meaning would have had little to do with our modern sense of democracy. Instead, the original meaning would have referred to those of high social rank (who were elected by fate, perhaps, to play that high born role), or those ministers of state who were favoured by the king, or a figure like the Pope who is (ultimately though indirectly) chosen by God. In fact, it was the transition to democratic modes of government in recent centuries that gave the word 'elite' a tarnished reputation. The 'elite' were precisely those NOT chosen by the people: those who had power simply because of wealth or social position or historical accident.

Correspondingly, 'elitism' has two meanings. The first is a political theory that says rule by the most capable, wise and educated is a good thing. Such a theory can trace itself back to Plato's Republic. It doesn't matter who these people are. It is a historical accident that the people in power already were also the ones who got educated, and thus elite in the sense of 'high social rank' tended to overlap with elite in the sense of 'the best to lead'. The second meaning, however, is a description of a person or institution that acts in ways to reinforce the position of the few. Or, in other words, someone or something that doesn't understand, care for, or act in the interests of those who are not already in positions of interest, who are assumed to be in the majority. Some accuse certain universities of 'elitism', for example, in that for one reason or another they tend to accept students from already privileged backgrounds. Calling a politician 'elitist', then, is tantamount to saying he or she is not a 'man/woman of the people'.

Should our leaders be the 'elite'? Well, yes and no. Yes, in the sense that we all hope the democratic process works and thus genuinely capable people are given the job. No, in the sense that they are not chosen from above, by a higher authority, but 'from below' so to speak, by the people.

I don't think that there is or could be a general principle that says that a paradox arising from idealizations will inevitably carry over (much less become worse) when the idealizations are relaxed. In some cases, the paradox will disappear when the idealizations are removed. In other cases, the paradox will persist (or become worse). There is no general rule here. It depends entirely on the details of the case.

For example, various paradoxes result in classical electromagnetic theory when pointlike charged particles are used. Point charges are a convenient idealization for many purposes, but the energy in such a field is undefined (the integral blows up). However, if we go to charge densities and extended charged bodies rather than point charges, these problems disappear. Likewise, in cosmology, Newtonian gravitational theory is afflicted with various paradoxes if we assume an infinite universe with a homogeneous, isotopic distribution of matter. Remove these idealizations and the problems go away.

These are intended merely as examples to show that sometimes paradoxes arise because of the idealization. In other cases, they arise despite the idealization. The matter can only be resolved in a case-by-case way, given the details of the particular paradox.

I'd separate the question whether mathematical statements are (often) existential from the question of the status of set theory. (Sure, we can construct faithful proxies inside set theory for most of the structures that mathematicians are interested in. But it is a moot question whether this makes set theory foundational in any good sense at all.)

Now, many mathematical statements are pretty uncontroversially not existential, but have the form "if anything is A it is B". So the theorem that anything which is a finite division ring is commutative doesn't tell us that there are such things as finite division rings, but only what they must be like if they do exist.

But of course many other common or garden mathematical theorems certainly do look existential. "There are an infinite number of prime numbers" looks existential -- and it is naturally read as implying that there are prime numbers (lots of them!). "There are four regular star polyhedra" looks existential -- and it is naturally read as implying that there are regular star polyhedra. And so it goes.

Perhaps appearances are deceptive, however. Perhaps these superficially existential statements are really non-committal "if ... then ..." statements in disguise. So really what we are saying, for example, is something along the following lines: if any collectionof things has the natural-number structure, then it contains an infinite number of things filling the prime-number role -- leaving it open whether there is anything that satisfies the antecedent of the conditional. Alternatively, maybe "there are an infinite number of prime numbers" is a statement made inside an essentially fictional mode of discourse (the arithmetical fiction), and no more really implies that there are numbers than "Sherlock Holmes lived at 222B Baker Street" implies that there really was such-and-such a man living along Baker Street.

Now, "if ... then ..."-ism and fictionalism are problematic as stories about the status of mathematics. I'm not recommending either position! But they are enough to illustrate the point that it isn't, perhaps, so obvious after all that prima facie existential mathematical statements have to be construed as really being kosher existential statements.

Stewart Shapiro's Thinking about Mathematics gives a terrific introduction to some of the issues hereabouts.

I don't know what you mean by: "One can create axioms that make statements like 'all bachelors are married' true". I assume that by 'married' you mean 'unmarried'. But I still don't understand. Perhaps you mean that one can write down obviously true principles from which the truth of every analytic sentence, and no other sentence, follows. But we can't.


Not yet anyway.