The hidden half of logic: a blind-spot in the zeitgeist of relativism?

[PseuTonym]

Look at any argument that relies upon non-Euclidean geometry and that attempts to reach the conclusion that in mathematics we are restricted to working within some language and deducing, from some list of assumptions in that language, a conclusion in that language.

You will observe something very strange: after citing a particular example of mathematical knowledge, there is a claim that mathematical knowledge is restricted to a form that does not leave any room for that example!

In particular, the example of mathematical knowledge is:
it is not possible to deduce the parallel postulate from the other postulates of Euclidean geometry.

The hidden half of logic allows us to show that it is impossible to deduce some conclusion given some assumptions.

 

[Tomdstone]

it is not possible to deduce the parallel postulate from the other postulates of Euclidean geometry.

That is because the other postulates are not sufficient to give a complete description of plane geometry.

 

[PseuTonym]

That is because the other postulates are not sufficient to give a complete description of plane geometry.

Suppose that I directed you to a documentary movie about Thomas Edison, showing live-action video of him talking and playing as a young child. I am not talking about a re-enactment, but actual archival film footage.

You could explain everything shown on the film. The problem is not what is shown, but how it was possible for it to be captured on film. Who invented the motion picture camera that was used to film him as a child?

Similarly, for you the problem is: how is it possible to acquire mathematical knowledge that the parallel postulate cannot be deduced?

We have to distinguish between the failure during some historical era to determine whether or not some conjecture can be deduced, and the acquisition of knowledge that it is impossible for a conjecture to be deduced.

 

[inocente]

In particular, the example of mathematical knowledge is:
it is not possible to deduce the parallel postulate from the other postulates of Euclidean geometry.

The hidden half of logic allows us to show that it is impossible to deduce some conclusion given some assumptions.

Similarly, for you the problem is: how is it possible to acquire mathematical knowledge that the parallel postulate cannot be deduced?

We have to distinguish between the failure during some historical era to determine whether or not some conjecture can be deduced, and the acquisition of knowledge that it is impossible for a conjecture to be deduced.

For reference, Euclid’s five postulates:

To draw a straight line from any point to any point.
To produce a finite straight line continuously in a straight line.
To describe a circle with any center and radius.
That all right angles are equal to one another.
That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

As a non-mathematician, two things strike me.

None of the postulates are necessarily true. For instance the first wouldn’t apply in any world where space is discontinuous.

The other thing is that it seems obvious that the first four postulates are independent, that is, none of them can be deduced from the other three. But that’s not at all obvious with the fifth postulate (it’s hard for me even to visualize what it’s talking about), and apparently a number of mathematicians had to chip away at it across several centuries before finally proving that it couldn’t be deduced from the others.

Now, suppose that Martians are much better at geometry than humans, so much better that Marti, an eight year old Martian, only has to glance at the five postulates to know for certain that the fifth can’t be deduced from the others. How does she do that? Perhaps in the same way we look at the third postulate and see that the others don’t mention circles, so it must be independent. Perhaps it has to do with the kinds of descriptions we choose. Marti already has a concept (that different geometries are possible) which took us humans a long time to find and take seriously.

Oops, that’s relative .

 

[PseuTonym]

For reference, Euclid’s five postulates:

To draw a straight line from any point to any point.
[the other four edited out by PseuTonym]

As a non-mathematician, two things strike me.

None of the postulates are necessarily true. For instance the first wouldn’t apply in any world where space is discontinuous.

I am wondering why you are considering the question of whether or not a given statement is necessarily true.

I would think that it is obvious that whether or not a statement is true depends upon the meaning of the statement, and that the meaning of a statement depends upon the meanings of the terms that occur in the statement. Now, I agree that we can transform a statement via some syntactic manipulations permitted by deductive logic, and reach a conclusion while ignoring some meanings.

However, it is helpful to remember that we do have to pay attention to the meanings of the terms that we categorize as part of logic itself, terms such as “and”, “not”, “if … then”, etc. For example, although we can deduce P from (P and Q), we cannot replace “and” in (P and Q) with “or” and imagine that we are entitled to deduce P from (P or Q).

What kind of necessity do you have in mind when you ask yourself whether or not a given sentence is “necessarily true”?

I am wondering whether you might be hoping to take into consideration pragmatic plans about conjectures that we hope to prove. In that case, you might paint yourself into a corner that requires you to say that if a statement refers to particulars, and asserts no generalization, then it is neither “necessarily true” nor “not necessarily true.”

Suppose that we decide in advance upon a list of conjectures, and we consider it to be essential that each conjecture in the list will become either an official axiom or an official theorem. If we use that list as our basis for selecting the axioms, then whatever proofs we create are simply rationalizations, and it would be dishonest to claim that we believe the theorems because we deduced them from the axioms. No, we would really be basing everything upon a list of theorems that we have dictated or commanded as being what shall be deduced.

Do the following two statements mean anything to you?

1. “Nine is not prime” is not necessarily true because it wouldn’t apply in any world where three is not a whole number.
2. “Five is prime” is not necessarily true because it wouldn’t apply in any world where two and a half is a whole number.

 

[Tomdstone]

1. “Nine is not prime” is not necessarily true because it wouldn’t apply in any world where three is not a whole number.
2. “Five is prime” is not necessarily true because it wouldn’t apply in any world where two and a half is a whole number.

Give us an example of where two and a half is a whole number and/or where three is a number but not a whole number.

 

[inocente]

I am wondering why you are considering the question of whether or not a given statement is necessarily true.

I would think that it is obvious that whether or not a statement is true depends upon the meaning of the statement, and that the meaning of a statement depends upon the meanings of the terms that occur in the statement. Now, I agree that we can transform a statement via some syntactic manipulations permitted by deductive logic, and reach a conclusion while ignoring some meanings.

However, it is helpful to remember that we do have to pay attention to the meanings of the terms that we categorize as part of logic itself, terms such as “and”, “not”, “if … then”, etc. For example, although we can deduce P from (P and Q), we cannot replace “and” in (P and Q) with “or” and imagine that we are entitled to deduce P from (P or Q).

What kind of necessity do you have in mind when you ask yourself whether or not a given sentence is “necessarily true”?

I am wondering whether you might be hoping to take into consideration pragmatic plans about conjectures that we hope to prove. In that case, you might paint yourself into a corner that requires you to say that if a statement refers to particulars, and asserts no generalization, then it is neither “necessarily true” nor “not necessarily true.”

Suppose that we decide in advance upon a list of conjectures, and we consider it to be essential that each conjecture in the list will become either an official axiom or an official theorem. If we use that list as our basis for selecting the axioms, then whatever proofs we create are simply rationalizations, and it would be dishonest to claim that we believe the theorems because we deduced them from the axioms. No, we would really be basing everything upon a list of theorems that we have dictated or commanded as being what shall be deduced.

Do the following two statements mean anything to you?

1. “Nine is not prime” is not necessarily true because it wouldn’t apply in any world where three is not a whole number.
2. “Five is prime” is not necessarily true because it wouldn’t apply in any world where two and a half is a whole number.

It’s interesting that you didn’t comment on the main contention in my post, which was that what seems obvious to some may not be obvious to others, and instead you concentrated on a specific example of what seems obvious or not obvious.

As a non-mathematician, “three” is defined to be an integer, “nine” is defined as three threes, and “prime” as divisible only by itself and unity. Therefore “three is prime” is necessarily true, meaning it is true in all possible worlds by definition. If you like, the definitions in the English dictionary are the axioms, and “three is prime” follows as a correctly formed theorem. It asserts nothing about objective reality, in depends purely on the definitions (which, if arbitrarily altered would obviously affect the meaning of all sentences).

Whereas “To draw a straight line from any point to any point” does make an assertion - that in our world we can draw a line. But even in our world it’s only approximately true, as long as we don’t look too close at the line, since every line we draw must consist of atoms, and atoms are lumpy, so even our best “straight” line must in reality be bumpy.

 

[PseuTonym]

Give us an example of where two and a half is a whole number and/or where three is a number but not a whole number.

I am unable to provide an example, and the statements that you quoted do not mean anything to me. I think that counter-factual scenarios can be not only meaningful, but also important. For example, driving while intoxicated is a problem even if it happens only once per driver, and even if no traffic accident occurs for the vast majority of the drivers.

The problem is the phrase “necessarily true.” When step-by-step reasoning is true, the conclusion is sometimes said to be “necessarily” true, but the necessity is logical necessity relative to the assumptions used and the actual validity of the reasoning. In other words, if the assumptions are true, and the reasoning is valid, then the conclusion is true. In plain language, we are dealing simply with truth, and references to “possible worlds” (a concept from modal logic) are a distraction not warranted.

Of course, if somebody can explain what is meant, then I am ready and willing to attempt to be educated. I don’t claim that possible worlds are irrelevant. I simply fail to see the relevance. Furthermore, it eventually looks like little more than a knee-jerk reaction to divert the discussion into a dead end.

(I can imagine stage directions such as the following: … “Is number theory being mentioned? Okay, subvert the use of valid reasoning by changing the topic to possible worlds. Oh, the topic is political decision-making? Use logical fallacies that are easily shown to be fallacies via examples from elementary number theory. People are invoking number theory to demonstrate that fallacious reasoning is being relied upon? Change the topic back to possible worlds.”)

 

[Tomdstone]

I am unable to provide an example, and the statements that you quoted do not mean anything to me. I think that counter-factual scenarios can be not only meaningful, but also important. For example, driving while intoxicated is a problem even if it happens only once per driver, and even if no traffic accident occurs for the vast majority of the drivers.

The problem is the phrase “necessarily true.” When step-by-step reasoning is true, the conclusion is sometimes said to be “necessarily” true, but the necessity is logical necessity relative to the assumptions used and the actual validity of the reasoning. In other words, if the assumptions are true, and the reasoning is valid, then the conclusion is true. In plain language, we are dealing simply with truth, and references to “possible worlds” (a concept from modal logic) are a distraction not warranted.

Of course, if somebody can explain what is meant, then I am ready and willing to attempt to be educated. I don’t claim that possible worlds are irrelevant. I simply fail to see the relevance. Furthermore, it eventually looks like little more than a knee-jerk reaction to divert the discussion into a dead end.

(I can imagine stage directions such as the following: … “Is number theory being mentioned? Okay, subvert the use of valid reasoning by changing the topic to possible worlds. Oh, the topic is political decision-making? Use logical fallacies that are easily shown to be fallacies via examples from elementary number theory. People are invoking number theory to demonstrate that fallacious reasoning is being relied upon? Change the topic back to possible worlds.”)

To say that dividing any number by two will lead to a whole number means that according to that logic, a whole number can be infinitesimally small because you can continue dividing the number by two ad infinitum.