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You start with a set of definitions and "self evident" axioms. These cannot be proven or disproved.

Using the rules of mathematical logic you then deduce other statements theorems). If the axioms are true, then these theorems must also be true. You can then use the axioms and the theorems to derive more true statements and so on. Once proven, you can assume that they are true without having to go back to the axioms every time.

Euclid formalised geometry in this fashion and all was well until his parallel postulate (an axiom) was questioned. The original was phrased differently (and in Egyptian, I guess), but it can be paraphrased as follows:

"Given a straight line and a point outside the line, there is exactly one line that goes through the point and is parallel to the original line."

Mathematicians in the 19th century found that they could develop axiomatic geometries replacing this postulate with its two alternatives: no parallel lines or many parallel lines, along with the other Euclidean axioms. They found that these geometries were wholly consistent.

So, you could have a perfectly good axiomatic geometry with Euclid's parallel postulate as well as with its negations!

Bertrand Russell tried to do the same for mathematics but failed. Then, in 1931 Kurt Godel showed that Russell's project was doomed from the start. Godel's incompleteness theorem proved that any (non-trivial) axiomatic system that was capable of arithmetic had to have statements which could be true and false - both versions were valid within the system.

However, you would need to be working with mathematics at a very high level before you need to deal with the issue of incomleteness.

Q: How do you know that various mathematical statements are true?

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pascals theory

Longer than you or anyone else will live! Thanks to Godel, there are statements about mathematical systems such that neither they, not their negation, can ever be proven to be true. This allows a whole new family of mathematical thinking to develop.

Mathematical or logical statements. Such as: 5 > 7 or 3 is a factor of 93 or 6 = 12

No. If you work within its definitions and the rules of logic it is not flawed. There are mathematical statements that you cannot prove to be true or false (Godel's incompleteness theorem), but that is not a flaw.

Lots of statements about animals are true.

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postulate

pascals theory

Such a statement is called a theorem.true

No. According to Godel's incompleteness theorem, in any mathematical system there must be statements that cannot be proven to be true or false. You simply cannot know!

Longer than you or anyone else will live! Thanks to Godel, there are statements about mathematical systems such that neither they, not their negation, can ever be proven to be true. This allows a whole new family of mathematical thinking to develop.

Without know what statements you are referring to we cannot answer.

"In mathematics, a proof is a demonstration that if some fundamental statements (axioms) are assumed to be true, then some mathematical statement is necessarily true." (from Wikipedia)

They were built at various times throughout the Americas.

Mathematical or logical statements. Such as: 5 > 7 or 3 is a factor of 93 or 6 = 12

No. If you work within its definitions and the rules of logic it is not flawed. There are mathematical statements that you cannot prove to be true or false (Godel's incompleteness theorem), but that is not a flaw.

we use various theorems and laws to prove certain geometric statements are true

Identities are statements that are true for any number.