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It seems to me that there are two kinds of numbers: the kind that the concept of which we can grasp by imagining a case that instantiates the concept, and the kind that we cannot imagine. For example, we can grasp the concept of 1 by imagining one object. The same goes for 2, 3, 0.5 or 0, and pretty much all the most common numbers. But there is this second kind that we cannot imagine. For example, i (square root of -1) or '532,740,029'. It seems to me that nobody can really imagine what 532,740,029 objects or i object(you see, I don't even know whether I should put 'object' or 'objects' here or not because I don't know whether i is single or plural; I don't know what i is) are like. So, Q1) if I cannot imagine a case that instantiates concepts like '532,740,029', do I really know the concept, and if so, how do I know the concept? Q2) is there a fundamental difference between numbers whose instances I can imagine and those I cannot? (I lead towards there is no difference, but I don't know how to account for this at least seemingly existent difference with regards to human imagination)

I'd suggest that while there may be differences in how easy it is for us to "picture" or "imagine" different numbers, this isn't a difference in the numbers themselves; it's a rather variable fact about us. I can mentally picture 5 things with no trouble. If I try for ten, it's harder (I have to think of five pairs of things.) If I try for 100, it's pretty hopeless, though you might be better at it than me. But I'm pretty sure that there's no interesting mathematical difference behind that. I'm also pretty sure that I understand the number 100 quite well. I don't need to be able to imagine 100 things to be able to see that 2x2x5x5 is the prime factorization of 10, for example, nor to see that 100 is a perfect square.

But that may still be misleading. I have no idea offhand whether 532,740,029 is prime. But I know what it would mean for it to be prime -- or not prime. And in fact, a bit of googling for the right calculators tells me that

532,740,029 = 43 x 1621 x 7643

I can't verify that by doing the math in my head, though some people can. But once again, I think I understand it. And once again, any limitations I might have are facts about me, and not about the number itself.

Someone might say that I don't have a specific concept of 532,740,029 in the way that have a specific concept of 5, and in a sense that may be true. Someone might add that I, Allen Stairs, am not capable of having the same sort of concept of 532,740,029 that I have of 5. And that, again, is true in a sense. But again: that doesn't mark a distinction among kinds of numbers.

Now i, the root of -1, is in a somewhat different category. It's not a number we use to count or measure things. But then, anyone who went around trying to find 5i objects doesn't understand i. But we can define complex numbers such as 3+i4 as pairs of real numbers wth a certain arithmetic, and any middle school child can master the arithmetic. That arithmetic is an extension of ordinary arithmetic. And it's an arithmetic that has uses. For example: in quantum theory, states are vectors in vector spaces where the coefficients can be complex numbers. This has various useful consequences. We don't need to be able to picture i for all of this to be true.

There is a sort of scale of abstraction, of course. 0 is a number, and recognizing that called for an act of abstraction. Likewise the negative numbers; likewise non-integer rational numbers; likewise algebraic and non-algebraic real numbers; likewise complex numbers; likewise transfinite cardinals and ordinals. We can make distinctions among numbers in terms of how they are related to other sorts of numbers. But the distinctions here are a matter of internal features of the numbers themselves and not our psychology.

I hope that's at least somewhat useful!

In my opinion, one of the reasons that we argue around determinism is that it seems to have some disturbing implications with regards to fatalism: if determinism is true, then everything is predetermined since the origin of the universe. That is to say that given enough information about the state of the original universe, it is possible to 'calculate' what is going to happen thereafter, because determinism means everything is strictly causally determined by its prior events. And because of this, in a strictly deterministic universe, there is only one 'fate' for anyone, and the disturbing implication that seems to follow is that since there is one fate for me, there is not much point for me make any decisions, because I'm not really making decisions, as everything I will do, or want, is already determined. How might a compatibilist, who thinks that humans are still capable of free will and are capable of making decisions, refute the above argument for fatalism? p.s.: this is a follow-up from the question about compatibilism... Thanks for such a great answer, Prof. Maitzen! p.p.s.: I too definitely don't think indeterminism is the way to go... But I also don't find determinism a whole lot more acceptable either, probably because I'm misunderstanding it from a 17-year-old point of view...?

Thanks for following up. I'm pleased that you found my earlier answer helpful.

Above you wrote, "the disturbing implication that seems to follow is that since there is one fate for me, there is not much point for me make any decisions, because I'm not really making decisions, as everything I will do, or want, is already determined." I'd like to make two points in reply.

(1) It's crucial not to confuse determinism with "your fate." Your fate is supposed to be the fixed outcome that you'll encounter regardless of anything you do in the meantime. So, according to the story, Oedipus is fated to kill his father and marry his mother, regardless of any actions Oedipus takes beforehand, including any attempts he makes to avoid that fate. Determinism is, if anything, the opposite doctrine. According to determinism, whom you marry (if anyone) depends crucially on your actions beforehand: every link in the causal chain is essential, no link is superfluous, and those links include your carefully considered prior decisions. As I tell my students, "Fatalism says that your grade on the Final Exam is already fixed and therefore doesn't depend on whether you decide to study for it between now and then. Determinism says that your grade is not already fixed but depends crucially on whether you decide to study for it between now and then."

(2) Yes, according to determinism, your decisions are themselves determined by earlier events, but that doesn't imply that those decisions aren't your decisions: after all, who else made them? No one thinks that determinism has a mind. Determinism isn't an agent. It can't make decisions. It can't manipulate. Nevertheless, unfortunately, even professional philosophers who ought to know better sometimes write as if determinism were a puppeteer pulling our strings. I'm not a psychologist, but I suspect that this confusion may stem from our evolved tendency to treat all causes as if they were agents. Nor does determinism imply that your decisions aren't decisions (what else are they?) or that your decisions are already made before you make them (again, who else is making them before you do?).

My question arises in free will and compatibilism. Basically, according to the compatibilists, the actions driven by 'internal factors' can be considered as free. Is this truly the kind of free will that people want to establish in the first place? Isn't this more of a compromise, rather than solution? I would have thought that the free will we are trying to seek is the capability to do otherwise, but I think internally driven actions are still determined, i.e. the agent could not have done otherwise. Moreover, is it right to seek free will as in 'the capability to do otherwise'? Is this truly meaningful? I feel like the whole deterministic and incompatible theory is somewhat dodgy in its logic: what does it mean that we cannot have done otherwise?

I would have thought that the free will we are trying to seek is the capability to do otherwise...

The capability to do otherwise, full stop, or the capability to do otherwise had we wanted to do otherwise? Today I saw my neighbor and gave him a friendly greeting because I wanted to. Even if determinism is true, had I wanted not to give him a friendly greeting -- imagine that he had rudely blasted his stereo and woken me at 3:00am -- there's no reason to think I would have given him a friendly greeting anyway.

That is, determinism is compatible with the claim that my desire to give my neighbor a friendly greeting played an essential role in my actually giving him a friendly greeting. Indeed, I want my action to be under the effective control of my desire: I don't want indeterminism to pop up in between the two. For suppose that, despite my wanting to give him a friendly greeting, something indeterministic popped up and resulted in my screaming obscenities at him instead. I would hardly be the model of a free and responsible agent; instead, I would be no more in control than someone suffering from Tourette Syndrome.

So determinism is compatible with our choosing as we desire to choose, which is the kind of freedom of choice that it makes sense for us to value. It makes no sense, I believe, to demand that we also be able to choose our desires. For one thing, if every choice is based at least partly on a desire, then given that we can't have made infinitely many prior choices, we must have at least some unchosen desires.

I don't see compatibilist freedom as any sort of compromise, because I don't see anything attractive in the alternative idea that, at the instant I make my choice, my desires and other facts about me don't determine which choice I'll make but, instead, leave room for something indeterministic to pop up and surprise me (and my neighbor).

My question regards the notion of negative rights. Personally, I believe the notion of “rights” is itself a human creation, and that rights do not ultimately exist outside of this creation. Rights come from nowhere else but humans. This being the case would seem to imply that all rights would, by definition, be positive, even if means determining a right not to do something. Humans desicion-making process itself entails deciding to do, or not to do something, or allowing, or not allowing something to be done, all of which have been positively decided. What am I not understanding?

As best I can tell, you may be confusing two different senses of the word "positive." When philosophers refer to your "positive" right, as opposed to your "negative" right, they typically mean your right to have some good or service provided to you, as opposed to your right not to be interfered with in some activity. So (putting it a bit simplistically perhaps) a right to adequate health care would be a positive right, while a right to speak freely in a public park would be a negative right.

But philosophers also use the word "positive" to label rights that are conferred by explicit human decrees, such as rights conferred on citizens by the decrees of legislatures or courts. The contrast is often with "natural" rights, which are supposed to be rights that we possess regardless of any human decree. You seem to be saying that all of our rights are positive rights in this second sense, which -- even if true -- wouldn't imply that all of our rights are positive rights in the first sense. A legislature or court can decree that you have the "negative" right to speak freely in a public park.

The Universal Declaration of Human Rights makes sweeping pronouncements about rights to housing, education, etc. Who is obliged, however, to see that such rights are positively protected or practiced--the universe? The UN has no institutional standing to require me to house the homeless. Isn't my role not to interfere with people seeking housing, just as I honor another' right to free speech by doing nothing to stifle her expression? The Declaration seems like it should be recast as a desired state document that serves as a guide for government policy and law. Do philosophers find the Declaration sound?

Your question actually seems to be several rolled into one.

In general, I think you're asking about what sorts of claims are being made when the UN Declaration states that there are rights to housing, education, etc. Your remark that you yourself only have to not interfere with others seeking housing seems reasonable (though I wonder whether you would agree that you have no obligation to provide housing if (say) your community is struck by a natural disaster that renders large numbers of people homeless). However, I gather that the UN Declaration is asserting that such rights are rights against one's state or society rather than against the members of states or societies. To say that individuals have a right to housing is not to say that particular individuals are obligated to house them. Instead, this right is one held against a collective: society, or the state as its representative.

Secondly, the rights to which you refer appear to be positive rather than negative rights. To have a negative right to X is to a have a moral claim that others not interfere with your pursuit or enjoyment of X. To have a positive right to X is to have a moral claim that others provide you with X. The UN Declaration is most plausibly interpreted as asserting that individuals have certain positive rights (housing and education among them) against their state or society. States or societies are obligated, on this interpretation, to provide such goods to their members. (Notice that these being positive rights against states or societies is compatible with such goods only grounding negative rights among individuals, i.e., individuals may not interfere with one another in the pursuit or enjoyment of these goods but do not have any obligations to provide such goods to one another.)

Lastly, you ask about the UN's standing to enforce these rights. Certainly there are important practical and political problems associated with the fact that the UN is not a 'world-state' with the capacity to enforce such provisions. But a more charitable interpretation of the UN Declaration is that it articulates moral standards to which the members states are to aspire. The rights in question are thus moral or "human" rights, rights individuals have 'naturally' and independently of their recognition by others, not legal rights enforced by some political body.

People often ask ill-conceived questions about whether, by using things like modern medicine, we obstruct natural selection. My own thought is that this thinking is facile, because it presumes that either medicine or caring for sick people are somehow "unnatural" interventions for our species. Animals like chimpanzees are known to use tools, and we wouldn't say that is unnatural for them--isn't medicine just a human tool? But then I wonder if conceiving of natural selection in this way makes it impossible to say that we ever obstruct natural selection, which seems like an logically odd consequence. I was wondering if there is a more discriminating formulation of natural selection according to which we obstruct natural selection in some cases and not others.

You're right to be suspicious of the idea that we somehow "obstruct natural selection." That way of putting things assumes a dubious notion of "natural" and suggests that if we influence evolution (which we do) this is somehow a bad thing. As you point out, medicine and other human inventions are natural in a perfectly good sense: we are part of nature and we create these things. Of course, there are also perfectly good uses of the word "natural" according to which medicines, computers, cars and so on aren't natural. But whether something is "natural" in that sense and whether it's good or bad are two very different questions. So is the question of whether these "unnatural" things have a place in evolution.

Some people think that because we can manipulate our environment, evolution doesn't apply to humans. This is confusion. Natural selection is a matter of differences in how likely an organism is to reproduce because of features of the environment it finds itself in. Even if we shape the environment, it's still true that in the environment we've partly shaped, people with some traits are more likely to reproduce and pass on their traits than others. And it isn't an all-or-nothing affair; it's a matter of more or less likely.

It's true that some traits are no longer important for selection. For example: in much of the world, being near-sighted is far less of a threat to reproductive success than it was centuries ago. This is because we invented glasses. On the other hand, we've created a world in which people who are better at multi-tasking have advantages that they may not have had two hundred years ago. This doesn't mean that bad multi-taskers will eventually die out, but it could still have an effect on differential rates of reproduction. We've also introduced toxins into the environment that humans once didn't have to contend with. This creates a selection pressure. If some people are less susceptible to those toxins, that could make them more likely to reproduce and pass their resistance along.

Principles used in evolutionary science such as the Hardy-Weinberg principle (http://anthro.palomar.edu/synthetic/synth_2.htm) don't distinguish between "natural" and "unnatural," and that's the key point. Evolutionary science isn't about a special set of environmental pressures that count as "natural." It's about how the distribution of heritable traits changes in a population because of how likely creatures with those traits are to reproduce, given the environment they find themselves in. I think you'd be hard-pressed to find an evolutionary scientist who took the idea of "obstructing natural selection" seriously.

Say I smoke marijuana not just because I find it pleasurable but because it is a form of counterculture expression. That is, smoking marijuana is a political act. I think that some in the 60s might have subscribed to that notion. I insist that weed is speech (and why not since money and flag burning seem to be), and the government cannot deny me exercise of that right. How do we square the general right of expression with problematic individual cases like the one above? Surely my exercise will get me arrested in many jurisdictions.

I have consulted an authoritative legal friend, who suggests the following line of thought. Is smoking marijuana a form of speech protected by the First Amendment? I cannot simply engage in an act and make the act a protected form of speech because I think or claim it is. There is also the objective test as to whether it has that meaning in the culture at large. In the 1960s smoking pot might have been taken as a protest, and yet today it might not, if only because its use is common. Besides, not any act, even literal speech, is protected under the First Amendment. One could not take unlawful killing, say, or "fighting words", for example incitements to criminal activity, to be protected. So far the law. It seems to me that very similar points apply, minus the Constitutional formulation, if the question is taken morally rather legally, in spite of the fact that the preservation of the democracy is not at the heart of morality.

Someone told me that there are nothing like table or chair because what there actually are is arrangement of matter and energy and label attached to it, which only exist in our minds. He thought if we accept nominalism, then we must accept this. I think he confused abstract objects with concrete objects. It seems to me that it's possible to believe things like table and chair exist while believe those concepts exist only in mind. Am I wrong?

Judging from your descriptions of them, your position seems to me more plausible than the other person's position.

If tables actually are arrangements of matter and energy to which some label (say, "table") attaches, how does that imply that tables don't exist? On the contrary, it seems to imply that tables do exist, because it implies that tables are things (namely, arrangements of matter and energy) to which that label attaches, i.e., things referred to by that label. Someone who denies the existence of tables ought not to identify tables as anything, including as particular arrangements of matter and energy.

It's true that tables wouldn't exist unless human beings (or some other species) made them: tables are artifacts. But of course it's not true that tables wouldn't exist unless human beings (or some other species) attached the label "table" to particular arrangements of matter and energy. Indeed, the very first tables probably weren't called "tables," and in most of the world tables still aren't called "tables."

Can tables exist even if the concept table exists only in our minds? I don't see why not. There's no reason to think that if a concept exists only in our minds, then whatever answers to that concept must also exist only in our minds. Similarly, just because all labels are items of language, it doesn't follow that everything that has a label is an item of language.

As far as I know, nominalists do not and need not deny the existence of tables or any other concrete entities. What nominalists are more likely to say is that if there were no minds, then nothing would have any properties: the earth wouldn't be spherical, for example. I reject that nominalist view, but even those who hold the view can agree with the rest of us that tables do exist.

Do we have moral duties towards institutions (like the Red Cross)? Do institutions have moral rights?

I often find the words "duty" and "rights" confusing outside of legal contexts, because they're weighted with theoretical overtones that don't always help us think clearly about how we should act and what we should do. So let me refocus the question: are the things we should and should't do when it comes to institutions? I think the answer is yes.

Suppose that I find a way to hack into the Red Cross bank accounts and steal money. I shouldn't do that. It's not just that it's against the law (though it certainly is). It's just wrong. It's not wrong just because it may hurt the CEO of the Red Cross, or any of the people who work for the Red Cross. Those people come and go, and it may even be that they aren't actually harmed by my act of theft. What I'm doing is wrong because (dare I say?) it harms the Red Cross itself. We could provide lots of related examples. And when it comes to the fundamental question, that's a pretty good way to answer it, I think. We can do things that help or harm organizations and institutions. Depending on the organization or institution's purposes, nature and so on, at least some of those are things we should or shouldn't do (say, not stealing from them) and some are things it might be good to do even if we aren't strictly obliged (say, donating money to them.) We could recast some of this in terms of rights and duties, and that might be just fine. I'm just skeptical about how much it will help to start with those more rarified concepts.

We also hold organizations responsible. It's true in law, of course: we can sue corporations. But we also make moral judgments. I might think that Doctors Without Borders is a commendable organization. I might think some shady organizations are despicable even if it they never break. the law. This reinforces the common-sense thought above that yes, there really are ways we should and shouldn't behave toward organizations.

We could have an interesting discussion about the metaphysics behind our moral attitudes here. I'd expect one of the conclusions to be that in some ways, organizations are a lot like persons. That's why in law we have a concept of legal personhood that includes things like corporations. I think we'd also come to the conclusion that persons are more like organizations tan we might have thought: they don't have some pure, unified metaphysical core. If that's right, it suggests that there's no good way to make a really sharp break between persons and organizations when we think about how we ought to act.

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