Ancient Philosophy and Contemporary Problems

Monday, February 27, 2006

In class, Professor Tillman said that while Euclidean geometry is mathematically consistent, it does not describe the way that the world really works. However, I think there is still a problem with the idea that mathematics does not necessarily describe the universe (or describe "the way things really are", which is actually an identical statement.) Namely, that another mathematically consistent set of ideas - non-Euclidean geometry (which in no way contradicts Euclidean geometry) - actually does describe the universe.

For any situation what so ever, one is able to find a particular mathematical reason for what is to be the way that it is - in fact, logic itself is a form of mathematics. Stating that there are certain branches of mathematics which do not actually describes the universe is equivalent to stating that there are certain logical systems which do not describe the universe; i.e. there are arguments which are valid but unsound.

I would view any mathematical theorem whatsoever as being merely a logical argument. In fact, the "proof" construct which is used in mathematics to provide theorems is identical to the "proof" construct used in logical philosophy to construct arguments. By all of this, I mean merely to say that if mathematics does not describe the universe, then it is necessary that logic does not either (since for any system of logic there can be constructed a corresponding system of mathematics.)

I hope my rant wasn't too lengthy; sorry for not formulating this post in the form of a formal argument.

I believe that I have found a major error in Max Black's paper. Either he doesn't understand the mathematics he discusses (which I view as being unlikely) or he was simply careless in some respect; however, he made a mistake in the first instance of his math that is so glaringly obvious I'm surprized that his editors didn't catch it!

The "convergent geometrical series" that he names doesn't converge to 11.1 - it can't! Why? Because his first term is 100 - which is greater than 11.1! Instead, here's a calculation of the value that the series converges to:

S=a/(1-r)
S=100/(1-1/10)
S=111.11111...

What I find to be really disturbing is that he keeps on naming 11.1 as the value at which the series converges - while high school math shows this to be incorrect. Also, he names 11.1 as one value to which his series apparently converges and 11 and 1/9 as another, which differ by 0.0111.... or 0.1111...% While I admit that this deviation is pretty small, it still doesn't change the fact that its quite obvious that 11 and 1/9 is a different number than 11.1. So, not only does he quote the wrong number as the value at which the series converges, but he also contradicts himself in naming the same number!

In case you were wondering, he makes the mistake right after he lists out the terms of the series. He says:

"So the total number of yards he must travel in order to catch the tortoise is

100+10+1+1/10+...

This is a convergent series whose sum can be expressed in decimal notation as 11.1, that is to say exactly 11 1/9."

Assignment for 2/29

Please read Max Black's article, "Achilles and the Tortoise". It is available from the course website. In it he claims (and subsequently argues for) the following:

"(Though a) mathematical solution (to Zeno's paradox of Achilles and the Tortoise) has behind it the authority of Descartes and Pierce and Whitehead--to mention no lesser names--. . . I cannot see that it goes to the heart of the matter. It tells us, correctly, when and where Achilles and the tortoise will meet, if they meet; but it fails to show that Zeno was wrong in claiming that they could not meet."

(Black 1951: 122; editing by Chris Tillman, emphasis in the original.)

Is Black's suggestion that a purely mathematical solution is inadequate correct? Why or why not? For next class, please briefly state and explain/defend your answer.

Assignment for 2/29

Please read Max Black's article, "Achilles and the Tortoise". It is available from the course website. In it he claims (and subsequently argues for) the following:

"(Though a) mathematical solution (to Zeno's paradox of Achilles and the Tortoise) has behind it the authority of Descartes and Pierce and Whitehead--to mention no lesser names--. . . I cannot see that it goes to the heart of the matter. It tells us, correctly, when and where Achilles and the tortoise will meet, if they meet; but it fails to show that Zeno was wrong in claiming that they could not meet."

(Black 1951: 122; editing by Chris Tillman, emphasis in the original.)

Is Black's suggestion that a purely mathematical solution is inadequate correct? Why or why not? For next class, please briefly state and explain/defend your answer.

Wednesday, February 22, 2006

Assignment for 2/27

Please extract an argument from Zeno and present it in numbered premise-conclusion form. State which premise is least plausible and why. Extra credit will be given to extractions of more difficult arguments.

Monday, February 20, 2006

Assignment for 2/22

For Wednesday, please read the Zeno fragments. (Pages 7-9 of Reading 2.) Also, please extract an argument from Aristotle against Parmenides. (Pages 8-11 of Reading 3. Do not use the same argument that you used for your paper.) Present the argument against Parmenides as a valid argument in numbered premise-conclusion form. State which premise you find least plausible and why you find that premise least plausible. Feel free to post your argument or any questions if you get stuck.

Parmenidean Arguments

Existence and Thought

1. There are no non-existent objects.
2. If (1), then there are no non-existent objects that have any properties.
3. Therefore, there are no non-existent objects that have any properties. (1,2 MP)
4. If (3), then whatever has any properties exists.
5. Therefore, whatever has any properties exists. (3,4 MP)
6. If (5), then if A has the property of being spoken or thought of, then A exists.
7. Therefore, if A has the property of being spoken or thought of, then A exists. (5,6 MP)
8. If (7), then if A can be spoken or thought of, then A exists.
9. Therefore, if A can be spoken or thought of, then A exists. (7,8 MP)
10. If it’s true that if A can be spoken or thought of, then A exists, then it’s true that if A does not exist, then A cannot be spoken or thought of. (Law of logic)
11. Therefore, if A does not exist, then A cannot be spoken or thought of. (9,10 MP)

No Generation Argument
12. If (11), then it’s not the case that what is came from what is not.

Explanation: Call what is ‘Gary’ and what is not ‘Bob’. Then the idea that Gary was generated/born/had a first moment is the idea that Gary came from Bob. But now suppose (11) is true. Then Bob is nothing; i.e., Bob is non-existent. So Bob cannot be spoken or thought about. So there is no proposition that Gary came from Bob. So it’s not the case that Gary came from Bob. So (12) is true.

13. Therefore, it’s not the case that what is came from what is not. (11, 12 MP)


Parallels to the No Generation Argument may be given for the conclusion that what is is indestructible.

No Change/Motion Argument
19. If what is changes, then what is goes from not being F to being F.
20. If what is goes from not being F to being F, then there is a state of affairs that what is is not F.
21. Therefore, if what is changes, then there is a state of affairs that what is is not F. (19, 20 HS)
22. If (11), then there is no state of affairs that what is is not F.
(Explanation: Call the fact or state of affairs of what is being F ‘Roy’. Then the thought that what is is not F is the thought that Roy does not exist. But by (11), it cannot be thought that Roy does not exist.)
23. Therefore, there is no state of affairs that what is is not F. (11, 22 MP)
24. Therefore, what is does not change. (21, 23 MT)
25. If what is does not change, then what is does not move.
26. Therefore, what is does not move. (24, 25 MP)

No Plurality Argument
27. If there is more than one thing, then something is true of one thing and not another.
(Explanation: Suppose that there are at least two things, A and B. If A is distinct from B, then A must have some property that B does not have. One plausible candidate for the difference is that A must be located where B is not. An even more plausible candidate is that A has the property of not being B. A must have this property, given our assumption. So (27) is true.)
28. If something is true of one thing and not another, then there is a state of affairs that something is not F.
29. Therefore, if there is more than one thing, then there is a state of affairs that something is not F. (27, 28 HS)
30. If (11), then there is no state of affairs that something is not F.
31. Therefore, there is no state of affairs that something is not F. (11, 30 MP)
32. Therefore, there is not more than one thing. (29, 31 MT)

Sunday, February 19, 2006

Clarification?

Hi, I need some help with clarifying how the essay should be formatted. It is probably written down on a sheet I can't find, which is why I hope/think that you, class, might be able to help me out.

Should I present ALL of my premises and conclusion in ONE paragraph towards the beginning, and then proceed to explain them in following paragraphs... OR.... should I present premise 1, dedicate a paragraph to explaining it, then present premise 2, dedicate a paragraph to explaining it... etc?

Thanks.

Friday, February 17, 2006

i'm reworking my argument for melissus, and i just wanted to check that it is valid. i'd really appreciate any feedback.


1. If something exists now, then it must have come from something that existed already.
2. Something exists now.
3. Therefore, that something came from something else that existed already.

4. If something came from something else that existed already, then something cannot come from nothing.
5. Something came from something else that existed already.
6. Therefore, something cannot come from nothing.

7. If something cannot come from nothing, then there is no coming into existence.
8. Something cannot come from nothing.
9. Therefore, there is no coming into existence.

10. If there is no coming into existence, then something always was and always will be.
11. There is no coming into existence.
12. Something always was and always will be.

13. If something always was and always will be, then it is the same case for everything: everything that ever did exist always existed and always will exist.
14. Something always was and always will be.
15. Everything that ever did exist always existed and always will exist.


is it necessary to repeat the conclusions of each set as the second premise of the next set in order for the conclusion to be valid?

Wednesday, February 15, 2006

Could someone please email me or post the peer review sheet? I never recieved it. Thanks!

Writing Resources

I have created a page with writing resources that you should use in constructing your final prose draft of essay 1. The page is also available from the course website.

Sunday, February 12, 2006

The assignment is to present in numbered premise conclusion form an argument from Melissus on one of the following Parmenidean conclusions:

Everything that ever did exist always existed and always will exist.
There can never be more or fewere things than there ever were.
Nothing that lacks both a beginning and an end is unlimited.
There can only be one thing.
There is only one thing.
There is no motion.
Nothing has proper parts.
What exists is the same throughout.
There is no change.

This should be 3-4 pages in outline form with explanations and definitions of the premises. Next, present an objection to the argument from Melissus, using numbered premise-conclusion form. Again, explain the objection and define any technical terms. Then, reply to the objection and provide a reason for thinking that one of the premises is false. Finally, give an overall evaluation of the argument claiming whether or not it is sound. Don't forget to cite your sources.
Good luck!

i seem to have lost the sheet that says what to do for the outline due tomorrow. can someone give me a quick idea of what i should bring to class?

I was working on my argument for Melissus and I was wondering if this was valid:

(1) If it came to be, then before it came to be it was nothing. (EP)

(2) If [before it came to be] it was nothing, then it must have come to be out of nothing. (IP)

(3) It is not the case that anything could come to be out of nothing. (EP)

(4) So it is not the case that before it came to be it was nothing. (IP)

(5) So it is not the case that it came to be. (IP)

I know I can say that 4 follows from 2 and 3 by Modus Tollens. Can I also say that 5 follows by Modus Tollens from 1, 2 and 3? If I symbolize the argument it goes as follows:

If A, then B
If B, then C
It is never the case that C
So it is not the case that B
So it is not the case that A

Is this valid?

Wednesday, February 08, 2006

Aristotle and Pseudo-Aristotle readings are up on the course website. They are not pretty but they are readable. Additional copies are available at your friendly neighborhood library.

First Parmenidean Argument

1. There are no non-existent objects.
2. If (1) is true, then there are no non-existent objects that have properties.
3. Therefore, there are no non-existent objects that have properties.
4. If (3) is true, then whatever has any property exists.
5. Therefore, whatever has any property exists.
6. If (5) is true, then if A has the property of being spoken or thought of, then A exists.
7. Therefore, if A has the property of being spoken or thought of, then A exists.
8. If (7) is true, then if A can be spoken or thought of, then A exists.
9. Therefore, if A can be spoken or thought of, then A exists.

Here is the proper symbolization of (1-9):
1. A
2. A --> B
3. Therefore, B
4. B --> C
5. Therefore, C
6. C --> (D --> E)
7. Therefore, D --> E
8. (D --> E) --> (F --> E)
9. Therefore, F --> E

Here are explanations for each of the premises:

Explanation of (1): First a word about existence and identity. If A exists, then something is identical with A. In addition, if x = y, then x is F if, and only if, y is F. Finally, 'there are Fs' means that there exist Fs. Now for the reasoning in favor of (1). Suppose (1) is false. Then there exist objects that are non-existent. But that is impossible, since it is contradictory for there to exist objects that are non-existent. So (1) is true.

Explanation of (2): A property is any feature, attribute, or way that something is. Now note that (2) is a conditional ("if-then") premise. So, as with explaining all conditional premises, we must assume the "if"-part and give a reason for accepting the "then"-part. So assume that (1) is true. Then there exist no objects that are non-existent. But if that's so, then there exist no objects that are non-existent that have properties. (Compare: if there are no dogs, then it follows that there are no dogs that have fleas. If I have no dinner, then I have no steak dinner. Etc.) So (2) is true.

Explanation of (3): (3) follows from (1) and (2) by Modus Ponens.

Explanation of (4): (4) is another conditional premise. So assume that (3) is true. Suppose now that A has properties. Then either A exists or A does not exist. If A does not exist, then there is a non-existent object that has properties. But this contradicts our assumption, (3). So A exists. But there is nothing special about A. The same reasoning applies to any other thing. So (4) is true.

Explanation of (5): (5) follows from (4) and (3) by Modus Ponens.

Explanation of (6): Assume (5) is true. Assume A has the property of being spoken or thought of. Then A must exist; otherwise, we contradict our assumptions. So (6) is true.

Explanation of (7): (7) follows from (5) and (6) by Modus Ponens.

Explanation of (8): Assume (7) is true. Assume A can be spoken or thought of. Then A has the following property: being something that can be spoken or thought of. But then, by (7), it follows that A exists. So (8) is true.

Explanation of (9): (9) follows from (7) and (8) by Modus Ponens.

Tuesday, February 07, 2006

I think Parmenides’ argument can be disproved because it leads to a contradiction:

1. If A can be spoken or thought of, then either A possibly exists or A exists.
2. If either A exists or A possibly exists, then A exists.
3. Therefore, if A is spoken or thought of, then A exists.
4. If A exists, then A is real.
4. I am thinking about Mickey Mouse.
5. Mickey Mouse isn’t real.
6. If I am thinking about Mickey Mouse then Mickey Mouse is a member of the set of possible A’s.
6. Therefore, Mickey Mouse both exists and doesn’t exist.

Is this argument valid? Furthermore, would Parmenides have agreed with the premises? If Parmenides wouldn't have agreed with the premises, then I think it would have to be because of some misunderstanding of what is it was that Parmenides meant by the idea that we have an inability to speak of or think about things that don't exist. Obviously, people can lie, in which case they would be saying something that isn't true, and necessarily doesn't exist. Also, if someone can think about or talk about the falsehood of Parmenides ideas, then that's also a contradiction. If Parmenides' ideas were true, it would be impossible to discuss their falsehood, since their falsehood wouldn't exist! I think, therefore, that it would be important to discuss what is meant by "existing" - that is, what is the definition of "existence." I think that only if a universal definition of "existence" can be reached can Parmenides ideas be viewed as either true or false (because, as I believe I have shown above, certain definitions of existence lead to contradictions, while others, presumably, do not.)

Monday, February 06, 2006

Ancient Philosophy and Contemporary Problems

The last post was published on 2/5
This post suggests the next class is 2/8
Therefore, there is no class on 2/6

is this a valid conclusion, or am I mistaken?

Sunday, February 05, 2006

For next class (Wednesday 2/8) please workshop your peer's Parmenides argument. This involves doing the following:

1. Symbolize your peer's argument. (Refer to the argument handbook if you're not sure how to do this.)

2. Determine whether your peer's argument is formally valid.

3. Determine whether you think Parmenides would accept the premises that your peer attributes to him.

4. On the basis of (1-3), give suggestions to your peer for improving his/her argument.

Wednesday, February 01, 2006

Proof vs. Justification

I would like to try to clarify a crucial point that came up in class today. Consider the following argument for an anti-Thalean conclusion:

1. Water is H2O.
2. Some things are not H2O.
3. Therefore, (1) and (2). (from 1,2 by Conjunction)
4. If (3), then not everything is water.
5. Therefore, not everything is water. (from 3,4 by Modus Ponens)

(1-5) is a valid argument. Its premises are plausible--we do not have good reason to doubt them. So we are plausibly justified in believing that they are true. But since the argument is valid, we are plausibly justified in believing that the conclusion is true as well. So barring a good argument that one of the premises in (1-5) is false, we are plausibly justified in accepting the conclusion. So we should accept the conclusion.

My point in class is that, though I believe the foregoing is correct, we should not believe either (a) that (1-5) is a *proof* of its conclusion, in the mathematical or strictly logical sense, or (b) that the argument is no good unless Thales (or a Thalean) would be *forced* to accept all of its premises. Anyone can be so stubborn so as to resist a premise. But they are not playing the same game we are. The game is not to force others to believe so and so; rather, it is to figure out what we are plausibly justified in believing, given our evidence.

Some further comments are in order.

First, even if we accept (1-5), we are not entitled to *ignore* those who do not. Recall that it's plausible to accept the premises in (1-5) provided we lack good reasons to doubt them. Now if Thales, or a Thalean, comes along and gives a valid argument for the conclusion that some premise in (1-5) is false, and backs up the premises with reasons, then we no longer lack good reasons to doubt the premise. Thus, we should either no longer accept the premise in question or we should find what is wrong with the objection to it. A nice thing about validity is that we know that if the conclusion of the objection is false, one of the premises must be false as well. Takeaway point: THe goal of rational inquiry is to determine what is most plausible. The goal is not to make those who disagree cry "Uncle".

Second, different people are justified in believing different things. You have a justified belief about the color of your underwear, and I do not have a similarly justified belief, since you have evidence about the color of your underwear that I lack. What is reasonable to believe is what is best supported by your evidence. That evidence is different for different people. So two people can reasonably believe different things. But when the question is important for rational inquiry, our goal is to find out what is correct. The best way to proceed in figuring out what is correct is to consider arguments. Those for whom the evidence supports answer A should give their reasons (arguments) for thinking that A is the case. Those for whom the evidence supports answer B should give their reasons (arguments) for thinking B is the case. If it cannot be the case that both A and B are correct, the arguments will help us sort out whether it is more plausible that A is the case or that B is the case. Takeaway point: Rational inquiry is about sharing evidence to better justify our beliefs in answers to the questions that we are interested in (be they scientific, philosophical, or what have you).

Third (and finally!), given the above, it is hopefully clear that the following objection is no good:

Argument A does not prove its conclusion. Therefore, argument A is no good.

In the mathematical/logical sense of "proof", there is precious little that can be proven. (This is so despite the fact that infinitely many propositions can be proven in the strict mathematical/logical sense! What I mean is that few of the conclusions that we encounter on a day to day basis or in science or philosophy admit of anything like mathematical/logical proof.) In the strict sense of "proof", no one can prove that you have a face or that the sun will come out tomorrow. But that no one can prove such things is pretty clearly a poor reason for you to believe that you have no face and that the sun will not come out tomorrow.

I apologize about the length of this post; I think these issues are of vital importance, however. While the picture of rational inquiry I have outlined might be wrong in detail, and is certainly debatable (recall that we can rationally inquire about rational inquiry itself!), I believe it is substantially correct. Please feel free to comment about any aspect that you agree with/disagree with/find unclear/etc.

Today (2/1) we considered an objection to Thales on behalf of Anaximander. We thought it might work pretty well against the view that everything is made of water but not so well against the view that everything comes from water. Anaximander seems to have thought that the argument against Thales was sound and that his view--that everything came from something with no definite properties--avoids the objection he raised against Thales as well as relevantly similar objections. Is Anaximander right about this? Why or why not?

You are encouraged, but not required, to present and explain a valid argument for your conclusion. If you opt not to present and explain a valid argument, you must write a paragraph or two stating and defending your answer. You need not write more than that.

For next class, please read the Melissus fragments.

Also, please feel free to post or comment on any of the issues raised in class today. That is in large part what the blog is for. Contrary to popular belief, philosophy is not best done in an attic by one's self. It is best done when there is an open exchange of ideas. (If you need a less idealistic motivation, remember as well that you are required to make a certain number of substantial posts/comments and that those who make more than the required amount will be rewarded for their efforts.)