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Hi, I've been reading about transfinite cardinal numbers and was wondering if you could answer this question. Supposedly the set of integers has the same cardinality as the set of even integers (both are countably infinite) since there exists a bijection between the two sets. But at the same time, doesn't there also exist a function between the set of even integers and the set of integers that is injective while NOT bijective (g(x) = x), since the image of f does not compose all of the integers (only the even ones)? To clarify, let f and g be functions from the set of EVEN integers to the set of ALL integers. Let f(x) = 1/2 x, and g(x) = x. Both are injective functions, but f is onto while g is not. So f is a bijection, while g is merely 1-to-1. Why, then, can I not say that the set of even integers and the set of all integers do NOT have the same cardinality since there exists some 1-1 function that is not onto? It seems like I should be able to draw this intuitive conclusion since g is 1-1, so for every element in the first set there exists a corresponding element in the second, but since g is not onto there are more elements in the second than in the first? Thanks!
Peter Smith
August 17, 2009
(changed August 17, 2009)
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The background issue here is: what's the best way of extending our talk of one set's being "larger" than another from the familiar case of finite sets to the infinite case?
Now, in the finite case, we can say that [L] the set A is larger than the set B if and only if there is a 1-1 mapping bet... Read more
Well, I am a math major. I am about to graduate, and I wish to attend graduate school in philosophy. I took one class in the philosophy of science. I know it is not enough, but I really have a deep passion for philosophy. I read alot on metaphysics, philosophy of mathematics and science by myself. It is to the point that I can understand much of the material in professional philosophical papers. I have a deep interest on the issue of ontological commitment to abstract objects, and the nature of the laws of nature. I really want to be a philosopher. What can I do?
Eddy Nahmias
August 17, 2009
(changed August 17, 2009)
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I suggest:
1. you talk to the philosophy professors at your school and ask them lots of questions. Hopefully, there is someone that who has a good sense of what it takes to get into grad school in philosophy, to succeed, and to get a job.
2. you explore websites at some PhD and MA programs.... Read more
How do you determine the panel? How do you determine who is and who is not a philosopher, more importantly? Isn't there such thing as a philosopher without a PhD and tenure, or whatever else holds this group of panelists together?
Alexander George
August 17, 2009
(changed August 17, 2009)
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Yes, of course - degrees, tenure, etc. are not necessary conditions for being a philosopher. There is no set of defining conditions for being a philosopher, any more than there is such a set for being a carpenter. Here, we look for people who know a bit about philosophy's history, its gr... Read more
If we can neither prove nor disprove the existence of a 'God', is it rational to even consider the possibility that he/she exists? Without the dedication of the few who preach from the worlds' religious houses, the notion of a 'God' surely wouldn't cross the mind of even the most imaginative of thinkers?
Peter Smith
August 15, 2009
(changed August 15, 2009)
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We seem naturally to be prone to over-interpret our environment and to see natural events as the results of intelligent agents at work. And you can see why our evolutionary history should have led to this cast of mind: it was much better for our ancestors to be too quick to diagnose potential a... Read more
If we can neither prove nor disprove the existence of a 'God', is it rational to even consider the possibility that he/she exists? Without the dedication of the few who preach from the worlds' religious houses, the notion of a 'God' surely wouldn't cross the mind of even the most imaginative of thinkers?
Peter Smith
August 15, 2009
(changed August 15, 2009)
Permalink
We seem naturally to be prone to over-interpret our environment and to see natural events as the results of intelligent agents at work. And you can see why our evolutionary history should have led to this cast of mind: it was much better for our ancestors to be too quick to diagnose potential a... Read more
Why is the use of police force justifiable to stop the attempted murder of a neighbor, but military force unjustifiable when used to stop the attempted murders of civilians in other countries who peacefully advocate for human rights (speech, assembly, voting, etc.)?
Thomas Pogge
August 14, 2009
(changed August 14, 2009)
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There are various potentially relevant differences. First, interventions abroad are often more likely to be counterproductive. The foreign government committing or condoning the human rights violations may be so powerful that the attempt to stop it will cause much more death and destruction th... Read more
My question has to do with why we attribute differing levels of blame for equivalent actions that produce different outcomes. For example, why does attempted murder go along with a much lesser sentence than actual murder. If I try my hardest to kill you and you get away, how is that action less punishable than if I was sucessful? Another example would be drunk driving. Consider three drivers driving home after one too many drinks: One arrives home safely. Although there is a spectrum on this issue, most will consider this a minor offense. The next driver was pulled over and arrested. He goes to jail and is considered by most to be a criminal. The last driver hits and kills a pedestrian and is charged with Vehicular Homicide. When he eventually gets out of jail, this person is a pariah. Given that these three drivers had the same number of drinks in the same amount of time, how are any of these three situations different? From a judicial standpoint I can see an argument for the differing levels of punishment, but why is my (and I'd wager to say most people's) opinion of the three hypothetical drivers different in each case? It seems like their outcomes are based on circumstance and not any additional wrongdoing.
Thomas Pogge
August 14, 2009
(changed August 14, 2009)
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May I refer you to earlier discussion of this issue under questions 348 and 897? If you don't feel satisfied with the answers there, do feel free to ask a more specific further question.
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Do numbers exist?
Peter Smith
August 13, 2009
(changed August 13, 2009)
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Here's a simple argument. (1) There are four prime numbers between 10 and 20. (2) But if there are four prime numbers between 10 and 20, then there certainly are prime numbers. (3) And if there are prime numbers, then there must indeed be numbers. Hence, from (1) to (3) we can conclude that (4)... Read more
If we can neither prove nor disprove the existence of a 'God', is it rational to even consider the possibility that he/she exists? Without the dedication of the few who preach from the worlds' religious houses, the notion of a 'God' surely wouldn't cross the mind of even the most imaginative of thinkers?
Peter Smith
August 15, 2009
(changed August 15, 2009)
Permalink
We seem naturally to be prone to over-interpret our environment and to see natural events as the results of intelligent agents at work. And you can see why our evolutionary history should have led to this cast of mind: it was much better for our ancestors to be too quick to diagnose potential a... Read more
Occam's razor seems to be a devestating weapon when it comes to the atheist's argument to why God doesn't exist, or more precisely it is more likely that he doesn't exist. It seems the scientific community has the consensus that they will never rely on "God" for the answer to any problem. Will there ever be a scenario in which God might prove simpler than the scientific or mathmatical explanation, and Occam's razor can be used to justify a belief in God?
Eddy Nahmias
August 13, 2009
(changed August 13, 2009)
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What if a burning bush appeared out of nowhere in Central Park in front of hundreds of people and said, "It is God speaking. I have come to tell you that you should believe that I exist. Also, love one another. And, by the way, the health care bill does not say anything about a 'death panel'... Read more