Con MPP Rick Nicholls; maybe evolution should not be taught in schools

[Tecumsehsbones]

Well math is more logic--but logical induction and deduction themselves are on thin ice, epistemologically speaking. Physics and chemistry is all theories though. Working models. Actuall;y chemcistry is more a collection of heuristics.

Nope, math is just a theory. Take for example, the old 2+2=4. All you can truly say is that in all observed cases, 2+2=4. You cannot say that 2+2 will never equal 5.

 

[Zipperfish]

Nope, math is just a theory. Take for example, the old 2+2=4. All you can truly say is that in all observed cases, 2+2=4. You cannot say that 2+2 will never equal 5.

It's logical deduction. All statements in mathematics are derived from axioms. 2+2=4 is a theorem, not a theory.

Theorem - Wikipedia, the free encyclopedia

In mathematics, a theorem is a statement that has been proven on the basis of previously established
statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a
logical argument for the theorem statement given in accord with the rules of a
deductive system. The proof of a theorem is often
interpreted as justification of the truth of the theorem statement. In light of
the requirement that theorems be proved, the concept of a theorem is
fundamentally deductive, in contrast to the notion of a
scientific theory, which is empirical.[2]

 

[Tecumsehsbones]

It's logical deduction. All statements in mathematics are derived from axioms. 2+2=4 is a theorem, not a theory.

Theorem - Wikipedia, the free encyclopedia

I'll accept that. Math is an internally consistent system, at least at the lower levels. Whether or not that system correlates to reality is another question entirely.

 

[Dexter Sinister]

Nope, math is just a theory. Take for example, the old 2+2=4. All you can truly say is that in all observed cases, 2+2=4. You cannot say that 2+2 will never equal 5.

Actually, sometimes it does, you can even get 2+2=6, "for large values of 2,". as it's usually put. I've just tried this with a spreadsheet, and also got it to display 3+3=5. Only a calculator or computer can get answers like that, depending on how they shift between internal storage formats and external display formats, like whether they truncate or round the numbers, but that kind of thing really just means you've made a mistake.

More realistically, however, you *can* say that 2+2 will never equal 5. I'll spare you the esoteric details, but it's rooted in how mathematics defines numbers and permitted operations with them, and amounts to a statement like 2+2=4 by definition.

I'll accept that. Math is an internally consistent system, at least at the lower levels.

'Fraid not. Look up Godel's Theorem. Any system of mathematics complex enough to contain operations like addition and subtraction will contain propositions whose truth or falsity cannot be determined.

 

[Zipperfish]

I'll accept that. Math is an internally consistent system, at least at the lower levels. Whether or not that system correlates to reality is another question entirely.

Almost there. Unfortunately, using impeccable and absolutely beautiful logic, Godel showed that any non-trivial axiomatic system (of which math and logic are examples) is either inconsistent or incomplete. By incomplete, meaning that there are truths that cannot shown to be true (e.g. are not proveable) within the system.

"Godel Escher Bach" is a great book on the subject oif you are interested.

The proof of Gödel's Incompleteness Theorem is so simple, and so sneaky, that it is almost embarrassing to relate. His basic procedure is as follows:

1.Someone introduces Gödel to a UTM, a machine that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all.

2.Gödel asks for the program and the circuit design of the UTM. The program may be complicated, but it can only be finitely long. Call the program P(UTM) for Program of the Universal Truth Machine.

3.Smiling a little, Gödel writes out the following sentence: "The machine constructed on the basis of the program P(UTM) will never say that this sentence is true." Call this sentence G for Gödel. Note that G is equivalent to: "UTM will never say G is true."

4.Now Gödel laughs his high laugh and asks UTM whether G is true or not.

5.If UTM says G is true, then "UTM will never say G is true" is false. If "UTM will never say G is true" is false, then G is false (since G = "UTM will never say G is true"). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements.

6.We have established that UTM will never say G is true. So "UTM will never say G is true" is in fact a true statement. So G is true (since G = "UTM will never say G is true").

"I know a truth that UTM can never utter," Gödel says.
"I know that G is true. UTM is not truly universal."

 

[Dexter Sinister]

Excellent post there Zip, same basic message as mine but more erudite. I tip my hat to you.