What’s not to understand? These are very famous philosophical paradoxes, in Lego(r) form.

Another from The Daily Nous. Don’t worry, they’re not taking over or anything.

To me, the most familiar of these oldies but goodies (all pre-Socratic?) is Sorites , aka The Heap as our own post title puts it. Dion and Theon is new to me.

I think the cartoonist is going for modern uses and restatements of these, not so much their original enunciations. Like, back in the day, they didn’t talk about qualia.

Really? The best-known one to me is clearly the Ship of Theseus. Of course, my training is in biology, and the fact that an organism is an epiphenomenon, not an object, is something we probably all think about. (This probably applies especially to me, son of a philosophy major.)

As for qualia, they aren’t necessarily all ancient, just all philosophical.

A bottle of qualia is still a bit of a surprise.

There was a humorous series of “dining reviews” of the cafes and “canteens” at various physics research centres, by a travelling scientist. He had pictures of menu boards from several European facilities where you could order Quark.

For me, this is an atypical instance where “I don’t understand” does not equate to “I don’t get the joke”. I am familiar with “The Ship of Thesus”, but the other ones are new to me. It’s suffice to say I don’t really understand “Phenomenal Sorites” based on what little exposure I have from just this panel, but I understand the intent of the comic as a whole.

I thought the “Ship of Theseus” was easier to understand in the “Granddad’s Hammer” version. As in, “This is my Granddad’s hammer. It’s on its 5th handle and 2nd head since he gave it to me.”

@padraig Other versions of “Granddad’s Hammer” involve tour guides showing you Washington’s axe at Mt Vernon, or Lincoln’s axe at New Salem. I agree — with just two pieces, these make the point easier.

Well, they make it easier, but especially easier for those on the negative side – “It’s utterly not the same hammer! Even when just one replacement had been done, it was half a different substance, i.e. not the same thing.” But with hundreds of small parts, the argument is more insidious.

The “Ship of Theseus” is one of my favorite paradoxes.

And according to Wikipedia, there’s a real-world example:

“A literal example of a Ship of Theseus is DSV Alvin, a submersible that has retained its identity despite all of its components being replaced at least once.”

All your cells get replaced over time. Some cells, like the lining of your stomach, last only 3 days before getting replaced. Some cells last for years. But even your brain cells get replaced after about 30 years or so.
So are you the same person you were?

Another interesting thing. I was listening to a series of philosophy lectures, and the lecturer asked: how do you know you aren’t just a brain in a vat, with wires connected up to sensors? Would life be different for you if you were just a brain in a vat?

If I had had the opportunity I would have pointed out that I AM a brain in a vat. The vat is called my “skull”. Wires that we call nerves run out to my eyes and my ears and to the skin all over me. If you were to cut the eye wires I would go blind; cut the ear wires and I would go deaf. The brain directly experiences NOTHING at all. It doesn’t even feel pain if you poke it.

I get along just fine being a brain in a vat, experiencing the world only through moderate-fidelity connections to my sensory organs.

Speaking of ships, I wonder how much of “Old Ironsides’, the USS Constitution, is original. It’s been rebuilt a few times over the years.

The issue of originality has been discussed in railroad preservation forums as it relates to restoring locomotives, and more than one high-dollar automobile restoration has been debated involving the remains of a particular car that’s little more than maybe a chassis, some mangled, rusted sheetmetal and a serial number.

The comic didn’t really resonate with me. I wonder if the author found a LEGO (R) modeling program and tried to work a joke around it.

Theon and Dion are bosonic.

All your cells get replaced over time.

Some cells are never replaced, notably many brain cells.

However, for any metabolically active cell, all (or almost all) the atoms still get replaced.

My chromium supplement didn’t arrive on time, so I ended up gnawing on an auto grille!

I never thought about it this way before, but the “Is the Star Trek teleporter a murder machine?” question is a variant of the Ship of Theseus.

Mark In Boston: But in context, the “are you a brain in a vat?” question is usually understood as meaning “Are the inputs being fed into your brain occurring in the naturalistic, logical way that they appear to be coming in from an external world, or is some other entity running a simulation that they would be free to change at will?”

The Trek transporter can and has duplicated people and in one case combined two people into one.

Not for the first time, I find myself wondering how many philosophers really understand Gödel’s theorem. Or, for that matter, integral calculus.

I don’t think Gödel’s theorem is particularly difficult for philosophers to understand. If you can understand Turing’s proof that it no program can exist that can decide, for every possible program, whether that program will terminate and return a correct result, you can understand Gödel’s theorem.

If you find a philosopher who can’t, just tell him “Everything I say to you is a lie” and keep him busy for days.

Well, I wouldn’t think so either, and then the Stanford page on sorites is full of accounts of doomed attempts to reconcile vague propositions with binary logic.

Which is why, when you want to use Mathematical Induction in a proof, you must first subordinately prove that the property P(x) is preserved under incrementing the argument. In the Sorites as paradox presentations we are asked to assent without argument to the premise “if NotAHeap(n) then NotAHeap(n+1) for all n in PositiveInt”. (Using the pile-it-on version, not the take-it-down.)

Mark in Boston: Perhaps Gödel’s theorem shouldn’t be hard for philosophers to understand, but like quantum mechanics, it ends up getting used for a lot of nonsense that shows that the speaker either doesn’t understand it, or is happy to misuse it.

Mathematical induction can prove that all horses are the same color. Let n be the number of horses in a set, n >= 1. Obviously all horses in a set containing one horse are the same color, so the proposition is true for n = 1. Now assuming it is true for some value of n, take a set of n+1 horses. Take out one horse A. Now we have a set of n horses, all the same color. Put back horse A and take out horse B. Again we have a set of n horses, all the same color. And since A and B are each the same color as the other horses, all n + 1 horses are the same color.

Unfortunately there is a special case where n = 2 and the induction step fails.

Watch out for those special cases.

When I was back in school in the 90s (MS CS) I took a course in Graph Theory. Lots of proofs in that one. I hadn’t done a Proof by Induction in 20 years, so I had to get reacquainted with the techniques.

Well then, here’s a toughie for ya.

How do you know that Mathematical Induction is a valid proof form? Do we just accept, because it makes sense, that if we can prove a base case, and if we can prove a successor step, then we have proved the property or statement works for all n in the domain? Or is that something we can prove? Wouldn’t we need to prove the validity of Mathematical Induction before using it?

[Though of course in a class or textbook, you might want to have used it a bunch first, to motivate the value of looking into whether it is reliable.]
[Mark in Boston’s post about the horse of a different color of course just is having fun and shows that you need to take care in using it, not that it is invalid actually.]

(If it looks like I was making a fetish of always saying Mathematical Induction and not simply Induction, that’s because Induction in the broader sense is not universally trusted.)

Mitch4: “How do you know that Mathematical Induction is a valid proof form?”

Because it’s always worked in the past.

Mitch4: what’s not universally trusted? Induction over the natural numbers is a special case of induction on well-founded sets, and while it’s important that the sets are well-founded (that is, actually sets and not semantic hazmat) I’m not aware of other restrictions.

When you start trying to prove things about software, nearly every proof is by induction.

All odd numbers are prime:

•Mathematician: 3 is a prime, 5 is a prime, 7 is a prime, and by induction – every odd integer higher than 2 is a prime.
•Physicist: 3 is a prime, 5 is a prime, 7 is a prime, 9 is an experimental error, 11 is a prime. Just to be sure, try several randomly chosen numbers: 17 is a prime, 23 is a prime…
•Engineer: 3 is a prime, 5 is a prime, 7 is a prime, 9 is an approximation to a prime, 11 is a prime, 13 is a prime…
•Psychologist: 3 is a prime, 5 is a prime, 7 is a prime, 9 is a prime, 11 is a prime, 13 is a prime…

DaveInBoston: Sets and software are still mathematical induction. But more generally, “The sun has always risen in the East, therefore the sun will rise in the East tomorrow” is a form of induction that isn’t guaranteed to work.

In “What the Tortoise Said to Achilles” by Lewis Carroll, Achilles, who has finally caught Zeno’s Tortoise, has a discussion with it about the validity of syllogisms.

Consider the following:
A. “Things that are equal to the same are equal to each other.”
B. “The two sides of this triangle are things that are equal to the same.”
Z. “Therefore, the two sides of this triangle are things that are equal to each other.”

A and B are known to be true, and it is claimed that Z is true. But how is it that Z is true? Because if A and B are true, then Z must be true. But that itself is an assumption, and ought to be made explicit.

A. “Things that are equal to the same are equal to each other.”
B. “The two sides of this triangle are things that are equal to the same.”
C. “If A and B are true, then Z is true.”
Z. “Therefore, the two sides of this triangle are things that are equal to each other.”

Fine, but it doesn’t help. The argument is valid only if C is true. After all, if C is false, then no claim is made about Z.

D. “If A and B and C are true, then Z is true.”

Years later, Achilles is still writing, “HHHHHH: If A and B and C and D and E and F and G and H and I and J …”

You have to use well-defined terms. What does “equal to the same” mean? What does “equal” mean. What is this “same” that they are equal to?

Which is ultimately why, when you are doing Mathematical Logic, or Philosophical Logic, you have an object language and a metalanguage. An inference pattern, or axiom schema, or whatever plays that role in your system, is used in the meta language, not the object language.

The fun twist Carroll presents is that Modus Ponens (approximately) – the inference pattern they want to use – is being treated like part of the object language when stated as step C, but really belongs to the metalanguage.

What’s not to understand? These are very famous philosophical paradoxes, in Lego(r) form.

Another from

The Daily Nous. Don’t worry, they’re not taking over or anything.To me, the most familiar of these oldies but goodies (all pre-Socratic?) is

Sorites, akaThe Heapas our own post title puts it.Dion and Theonis new to me.I think the cartoonist is going for modern uses and restatements of these, not so much their original enunciations. Like, back in the day, they didn’t talk about

qualia.Really? The best-known one to me is clearly the Ship of Theseus. Of course, my training is in biology, and the fact that an organism is an epiphenomenon, not an object, is something we probably all think about. (This probably applies especially to me, son of a philosophy major.)

As for qualia, they aren’t necessarily all ancient, just all philosophical.

A bottle of qualia is still a bit of a surprise.

There was a humorous series of “dining reviews” of the cafes and “canteens” at various physics research centres, by a travelling scientist. He had pictures of menu boards from several European facilities where you could order

Quark.For me, this is an atypical instance where “I don’t understand” does not equate to “I don’t get the joke”. I am familiar with “The Ship of Thesus”, but the other ones are new to me. It’s suffice to say I don’t really understand “Phenomenal Sorites” based on what little exposure I have from just this panel, but I understand the intent of the comic as a whole.

I thought the “Ship of Theseus” was easier to understand in the “Granddad’s Hammer” version. As in, “This is my Granddad’s hammer. It’s on its 5th handle and 2nd head since he gave it to me.”

This might also qualify for synchronicity; see todays SMBC

https://www.smbc-comics.com/comic/race

@padraig Other versions of “Granddad’s Hammer” involve tour guides showing you Washington’s axe at Mt Vernon, or Lincoln’s axe at New Salem. I agree — with just two pieces, these make the point easier.

Well, they make it easier, but especially easier for those on the negative side – “It’s utterly not the same hammer! Even when just one replacement had been done, it was half a different substance, i.e. not the same thing.” But with hundreds of small parts, the argument is more insidious.

The problem of the Ship of Theseus is easily solved: https://existentialcomics.com/comic/121

A little synchronicity with Saturday Morning Breakfast Cereal

https://www.smbc-comics.com/index.php

Fridays Comic (4/30/21) The Race Problem.

The “Ship of Theseus” is one of my favorite paradoxes.

And according to Wikipedia, there’s a real-world example:

“A literal example of a Ship of Theseus is DSV Alvin, a submersible that has retained its identity despite all of its components being replaced at least once.”

It’s not quite a perfect Ship of Theseus, but I do like the Ise Jingu shrine, which is either 1500 years old, or 8 years old, depending on your perspective: https://knowledgestew.com/2016/04/japanese-shrine-rebuilt-every-20-years.html#:~:text=In%20a%20tradition%20that%20dates%20back%20to%20the,most%20sacred%20of%20the%20Shinto%20religion%20in%20Japan.

All your cells get replaced over time. Some cells, like the lining of your stomach, last only 3 days before getting replaced. Some cells last for years. But even your brain cells get replaced after about 30 years or so.

So are you the same person you were?

Another interesting thing. I was listening to a series of philosophy lectures, and the lecturer asked: how do you know you aren’t just a brain in a vat, with wires connected up to sensors? Would life be different for you if you were just a brain in a vat?

If I had had the opportunity I would have pointed out that I AM a brain in a vat. The vat is called my “skull”. Wires that we call nerves run out to my eyes and my ears and to the skin all over me. If you were to cut the eye wires I would go blind; cut the ear wires and I would go deaf. The brain directly experiences NOTHING at all. It doesn’t even feel pain if you poke it.

I get along just fine being a brain in a vat, experiencing the world only through moderate-fidelity connections to my sensory organs.

Speaking of ships, I wonder how much of “Old Ironsides’, the USS Constitution, is original. It’s been rebuilt a few times over the years.

The issue of originality has been discussed in railroad preservation forums as it relates to restoring locomotives, and more than one high-dollar automobile restoration has been debated involving the remains of a particular car that’s little more than maybe a chassis, some mangled, rusted sheetmetal and a serial number.

The comic didn’t really resonate with me. I wonder if the author found a LEGO (R) modeling program and tried to work a joke around it.

Theon and Dion are bosonic.

All your cells get replaced over time.Some cells are never replaced, notably many brain cells.

However, for any metabolically active cell, all (or almost all) the

atomsstill get replaced.My chromium supplement didn’t arrive on time, so I ended up gnawing on an auto grille!

I never thought about it this way before, but the “Is the Star Trek teleporter a murder machine?” question is a variant of the Ship of Theseus.

Mark In Boston: But in context, the “are you a brain in a vat?” question is usually understood as meaning “Are the inputs being fed into your brain occurring in the naturalistic, logical way that they appear to be coming in from an external world, or is some other entity running a simulation that they would be free to change at will?”

The Trek transporter can and has duplicated people and in one case combined two people into one.

And sometimes the results can be tragic!

https://www.imdb.com/title/tt0091064/mediaviewer/rm3093015809/

Not for the first time, I find myself wondering how many philosophers really understand Gödel’s theorem. Or, for that matter, integral calculus.

I don’t think Gödel’s theorem is particularly difficult for philosophers to understand. If you can understand Turing’s proof that it no program can exist that can decide, for every possible program, whether that program will terminate and return a correct result, you can understand Gödel’s theorem.

If you find a philosopher who can’t, just tell him “Everything I say to you is a lie” and keep him busy for days.

Well, I wouldn’t think so either, and then the Stanford page on sorites is full of accounts of doomed attempts to reconcile vague propositions with binary logic.

Which is why, when you want to use Mathematical Induction in a proof, you must first subordinately prove that the property P(x) is preserved under incrementing the argument. In the Sorites as paradox presentations we are asked to

assentwithout argument to the premise “if NotAHeap(n) then NotAHeap(n+1) for all n in PositiveInt”. (Using the pile-it-on version, not the take-it-down.)Mark in Boston: Perhaps Gödel’s theorem shouldn’t be hard for philosophers to understand, but like quantum mechanics, it ends up getting used for a lot of nonsense that shows that the speaker either doesn’t understand it, or is happy to misuse it.

Coincidentally just ran into this YouTube comment: https://www.youtube.com/watch?v=LIZwhWwSaJY&lc=Ugyoh_pe_5sCvCWw_Pd4AaABAg.9MtPM28Un3s9MtQLYuTZjn&ab_channel=HeterodoxIdeaswithCodeLaure : “Godel’s theorem is used a lot to say that science has failed and we can move to (insert pet ideology or program). Yeah, it is mischaracterized “

It can be hard to find a Lyotard with proper fit

Mathematical induction can prove that all horses are the same color. Let n be the number of horses in a set, n >= 1. Obviously all horses in a set containing one horse are the same color, so the proposition is true for n = 1. Now assuming it is true for some value of n, take a set of n+1 horses. Take out one horse A. Now we have a set of n horses, all the same color. Put back horse A and take out horse B. Again we have a set of n horses, all the same color. And since A and B are each the same color as the other horses, all n + 1 horses are the same color.

Unfortunately there is a special case where n = 2 and the induction step fails.

Watch out for those special cases.

When I was back in school in the 90s (MS CS) I took a course in Graph Theory. Lots of proofs in that one. I hadn’t done a Proof by Induction in 20 years, so I had to get reacquainted with the techniques.

Well then, here’s a toughie for ya.

How do you know that Mathematical Induction is a valid proof form? Do we just accept, because it makes sense, that if we can prove a base case, and if we can prove a successor step, then we have proved the property or statement works for all n in the domain? Or is that something we can

prove? Wouldn’t we need to prove the validity of Mathematical Induction before using it?[Though of course in a class or textbook, you might want to have used it a bunch first, to motivate the value of looking into whether it is reliable.]

[Mark in Boston’s post about the horse of a different color of course just is having fun and shows that you need to take care in using it, not that it is invalid actually.]

(If it looks like I was making a fetish of always saying Mathematical Induction and not simply Induction, that’s because Induction in the broader sense is

notuniversally trusted.)Mitch4: “How do you know that Mathematical Induction is a valid proof form?”

Because it’s always worked in the past.

Mitch4: what’s not universally trusted? Induction over the natural numbers is a special case of induction on well-founded sets, and while it’s important that the sets are well-founded (that is, actually sets and not semantic hazmat) I’m not aware of other restrictions.

When you start trying to prove things about software, nearly every proof is by induction.

All odd numbers are prime:

•Mathematician: 3 is a prime, 5 is a prime, 7 is a prime, and by induction – every odd integer higher than 2 is a prime.

•Physicist: 3 is a prime, 5 is a prime, 7 is a prime, 9 is an experimental error, 11 is a prime. Just to be sure, try several randomly chosen numbers: 17 is a prime, 23 is a prime…

•Engineer: 3 is a prime, 5 is a prime, 7 is a prime, 9 is an approximation to a prime, 11 is a prime, 13 is a prime…

•Psychologist: 3 is a prime, 5 is a prime, 7 is a prime, 9 is a prime, 11 is a prime, 13 is a prime…

DaveInBoston: Sets and software are still mathematical induction. But more generally, “The sun has always risen in the East, therefore the sun will rise in the East tomorrow” is a form of induction that isn’t guaranteed to work.

In “What the Tortoise Said to Achilles” by Lewis Carroll, Achilles, who has finally caught Zeno’s Tortoise, has a discussion with it about the validity of syllogisms.

Consider the following:

A. “Things that are equal to the same are equal to each other.”

B. “The two sides of this triangle are things that are equal to the same.”

Z. “Therefore, the two sides of this triangle are things that are equal to each other.”

A and B are known to be true, and it is claimed that Z is true. But how is it that Z is true? Because if A and B are true, then Z must be true. But that itself is an assumption, and ought to be made explicit.

A. “Things that are equal to the same are equal to each other.”

B. “The two sides of this triangle are things that are equal to the same.”

C. “If A and B are true, then Z is true.”

Z. “Therefore, the two sides of this triangle are things that are equal to each other.”

Fine, but it doesn’t help. The argument is valid only if C is true. After all, if C is false, then no claim is made about Z.

D. “If A and B and C are true, then Z is true.”

Years later, Achilles is still writing, “HHHHHH: If A and B and C and D and E and F and G and H and I and J …”

You have to use well-defined terms. What does “equal to the same” mean? What does “equal” mean. What is this “same” that they are equal to?

Which is ultimately why, when you are doing Mathematical Logic, or Philosophical Logic, you have an object language and a metalanguage. An inference pattern, or axiom schema, or whatever plays that role in your system, is used in the meta language, not the object language.

The fun twist Carroll presents is that Modus Ponens (approximately) – the inference pattern they want to use – is being treated like part of the object language when stated as step C, but really belongs to the metalanguage.