How do mathematical proofs work?
$begingroup$
How do mathematical "proofs" work? What exactly are you proving? Take for example the pythagorean theorem: a^2 + b^2 = c^2. I looked up some proofs for it and here is one article that gives proofs.
First off, I think the equation a^2 + b^2 = c^2 (and even all math equations) is an inferential statement; meaning it is just a statement written in symbols about the observation that "the square root of the sum of the square of the sides of a right triangle is equal to the length of the hypotenuse."
What I mean is someone kept measuring the sides of the right triangles, squared them, added them, then took the square root, and found out that it's equal to the length of the hypotenuse. This was always true, and then using mathematical symbols, they expressed that fact of observation.
For example, in the article I linked, part of the proof for method 1 says,
The length of each leg of the right triangle is the geometric mean of
the lengths of the hypotenuse and the segment of the hypotenuse that
is adjacent to the leg
But this also is another fact of observation. But the pythagorean equation itself is a fact of observation, so how can you prove one fact of observation using another fact of observation? You can just say the proof for the pythagorean theorem is that it always works in practice?
soft-question proof-explanation
edited Dec 17 '18 at 16:32
Mauro ALLEGRANZA
67.1k449115
asked Dec 17 '18 at 16:23
KingamereKingamere
1166
$endgroup$
|
show 9 more comments
$begingroup$
How do mathematical "proofs" work? What exactly are you proving? Take for example the pythagorean theorem: a^2 + b^2 = c^2. I looked up some proofs for it and here is one article that gives proofs.
First off, I think the equation a^2 + b^2 = c^2 (and even all math equations) is an inferential statement; meaning it is just a statement written in symbols about the observation that "the square root of the sum of the square of the sides of a right triangle is equal to the length of the hypotenuse."
What I mean is someone kept measuring the sides of the right triangles, squared them, added them, then took the square root, and found out that it's equal to the length of the hypotenuse. This was always true, and then using mathematical symbols, they expressed that fact of observation.
For example, in the article I linked, part of the proof for method 1 says,
The length of each leg of the right triangle is the geometric mean of
the lengths of the hypotenuse and the segment of the hypotenuse that
is adjacent to the leg
But this also is another fact of observation. But the pythagorean equation itself is a fact of observation, so how can you prove one fact of observation using another fact of observation? You can just say the proof for the pythagorean theorem is that it always works in practice?
soft-question proof-explanation
edited Dec 17 '18 at 16:32
Mauro ALLEGRANZA
67.1k449115
asked Dec 17 '18 at 16:23
KingamereKingamere
1166
$endgroup$
2
$begingroup$
"What exactly are you proving?" Mathematical statements, often written in symbolic form.
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:26
1
$begingroup$
Many mathematical statements, like e.g. arithmetical and geometrical ones are used in "everyday life" : to count colelctions of objects, to measure physical and "practical" magnitudes.
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:28
1
$begingroup$
A proof is the derivation of a statement from specific axioms (i.e. the axioms assumed as basic facts of the discipline involving the said statement) using rules of logic (and previously proved statements, called theorems).
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:29
3
$begingroup$
Not at all; the Pythagorean theorem has been proved e.g. by Euclid (ca.300 BCE, Ancient Greece) for geometrical axioms (see Euclid's Elements).
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:30
2
$begingroup$
It might help to look at an example of a mathematical proof before asking a question like this. If you google, for example, "proof of the Pythagorean theorem" you'll see some indication of what a mathematical proof consists of. It's not simply "we always see that it works."
$endgroup$
– Noah Schweber
Dec 17 '18 at 16:33
|
show 9 more comments
$begingroup$
How do mathematical "proofs" work? What exactly are you proving? Take for example the pythagorean theorem: a^2 + b^2 = c^2. I looked up some proofs for it and here is one article that gives proofs.
First off, I think the equation a^2 + b^2 = c^2 (and even all math equations) is an inferential statement; meaning it is just a statement written in symbols about the observation that "the square root of the sum of the square of the sides of a right triangle is equal to the length of the hypotenuse."
What I mean is someone kept measuring the sides of the right triangles, squared them, added them, then took the square root, and found out that it's equal to the length of the hypotenuse. This was always true, and then using mathematical symbols, they expressed that fact of observation.
For example, in the article I linked, part of the proof for method 1 says,
The length of each leg of the right triangle is the geometric mean of
the lengths of the hypotenuse and the segment of the hypotenuse that
is adjacent to the leg
But this also is another fact of observation. But the pythagorean equation itself is a fact of observation, so how can you prove one fact of observation using another fact of observation? You can just say the proof for the pythagorean theorem is that it always works in practice?
soft-question proof-explanation
edited Dec 17 '18 at 16:32
Mauro ALLEGRANZA
67.1k449115
asked Dec 17 '18 at 16:23
KingamereKingamere
1166
$endgroup$
How do mathematical "proofs" work? What exactly are you proving? Take for example the pythagorean theorem: a^2 + b^2 = c^2. I looked up some proofs for it and here is one article that gives proofs.
First off, I think the equation a^2 + b^2 = c^2 (and even all math equations) is an inferential statement; meaning it is just a statement written in symbols about the observation that "the square root of the sum of the square of the sides of a right triangle is equal to the length of the hypotenuse."
What I mean is someone kept measuring the sides of the right triangles, squared them, added them, then took the square root, and found out that it's equal to the length of the hypotenuse. This was always true, and then using mathematical symbols, they expressed that fact of observation.
For example, in the article I linked, part of the proof for method 1 says,
The length of each leg of the right triangle is the geometric mean of
the lengths of the hypotenuse and the segment of the hypotenuse that
is adjacent to the leg
But this also is another fact of observation. But the pythagorean equation itself is a fact of observation, so how can you prove one fact of observation using another fact of observation? You can just say the proof for the pythagorean theorem is that it always works in practice?
soft-question proof-explanation
soft-question proof-explanation
edited Dec 17 '18 at 16:32
Mauro ALLEGRANZA
67.1k449115
asked Dec 17 '18 at 16:23
KingamereKingamere
1166
edited Dec 17 '18 at 16:32
Mauro ALLEGRANZA
67.1k449115
asked Dec 17 '18 at 16:23
KingamereKingamere
1166
edited Dec 17 '18 at 16:32
Mauro ALLEGRANZA
67.1k449115
edited Dec 17 '18 at 16:32
Mauro ALLEGRANZA
67.1k449115
edited Dec 17 '18 at 16:32
Mauro ALLEGRANZA
67.1k449115
67.1k449115
asked Dec 17 '18 at 16:23
KingamereKingamere
1166
asked Dec 17 '18 at 16:23
KingamereKingamere
1166
asked Dec 17 '18 at 16:23
KingamereKingamere
1166
1166
2
$begingroup$
"What exactly are you proving?" Mathematical statements, often written in symbolic form.
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:26
1
$begingroup$
Many mathematical statements, like e.g. arithmetical and geometrical ones are used in "everyday life" : to count colelctions of objects, to measure physical and "practical" magnitudes.
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:28
1
$begingroup$
A proof is the derivation of a statement from specific axioms (i.e. the axioms assumed as basic facts of the discipline involving the said statement) using rules of logic (and previously proved statements, called theorems).
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:29
3
$begingroup$
Not at all; the Pythagorean theorem has been proved e.g. by Euclid (ca.300 BCE, Ancient Greece) for geometrical axioms (see Euclid's Elements).
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:30
2
$begingroup$
It might help to look at an example of a mathematical proof before asking a question like this. If you google, for example, "proof of the Pythagorean theorem" you'll see some indication of what a mathematical proof consists of. It's not simply "we always see that it works."
$endgroup$
– Noah Schweber
Dec 17 '18 at 16:33
|
show 9 more comments
2
$begingroup$
"What exactly are you proving?" Mathematical statements, often written in symbolic form.
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:26
1
$begingroup$
Many mathematical statements, like e.g. arithmetical and geometrical ones are used in "everyday life" : to count colelctions of objects, to measure physical and "practical" magnitudes.
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:28
1
$begingroup$
A proof is the derivation of a statement from specific axioms (i.e. the axioms assumed as basic facts of the discipline involving the said statement) using rules of logic (and previously proved statements, called theorems).
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:29
3
$begingroup$
Not at all; the Pythagorean theorem has been proved e.g. by Euclid (ca.300 BCE, Ancient Greece) for geometrical axioms (see Euclid's Elements).
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:30
2
$begingroup$
It might help to look at an example of a mathematical proof before asking a question like this. If you google, for example, "proof of the Pythagorean theorem" you'll see some indication of what a mathematical proof consists of. It's not simply "we always see that it works."
$endgroup$
– Noah Schweber
Dec 17 '18 at 16:33
2
2
$begingroup$
"What exactly are you proving?" Mathematical statements, often written in symbolic form.
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:26
$begingroup$
"What exactly are you proving?" Mathematical statements, often written in symbolic form.
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:26
1
1
$begingroup$
Many mathematical statements, like e.g. arithmetical and geometrical ones are used in "everyday life" : to count colelctions of objects, to measure physical and "practical" magnitudes.
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:28
$begingroup$
Many mathematical statements, like e.g. arithmetical and geometrical ones are used in "everyday life" : to count colelctions of objects, to measure physical and "practical" magnitudes.
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:28
1
1
$begingroup$
A proof is the derivation of a statement from specific axioms (i.e. the axioms assumed as basic facts of the discipline involving the said statement) using rules of logic (and previously proved statements, called theorems).
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:29
$begingroup$
A proof is the derivation of a statement from specific axioms (i.e. the axioms assumed as basic facts of the discipline involving the said statement) using rules of logic (and previously proved statements, called theorems).
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:29
3
3
$begingroup$
Not at all; the Pythagorean theorem has been proved e.g. by Euclid (ca.300 BCE, Ancient Greece) for geometrical axioms (see Euclid's Elements).
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:30
$begingroup$
Not at all; the Pythagorean theorem has been proved e.g. by Euclid (ca.300 BCE, Ancient Greece) for geometrical axioms (see Euclid's Elements).
$endgroup$
– Mauro ALLEGRANZA
Dec 17 '18 at 16:30
2
2
$begingroup$
It might help to look at an example of a mathematical proof before asking a question like this. If you google, for example, "proof of the Pythagorean theorem" you'll see some indication of what a mathematical proof consists of. It's not simply "we always see that it works."
$endgroup$
– Noah Schweber
Dec 17 '18 at 16:33
$begingroup$
It might help to look at an example of a mathematical proof before asking a question like this. If you google, for example, "proof of the Pythagorean theorem" you'll see some indication of what a mathematical proof consists of. It's not simply "we always see that it works."
$endgroup$
– Noah Schweber
Dec 17 '18 at 16:33
|
show 9 more comments
2 Answers
2
active
oldest
votes
$begingroup$
Here's some sketchy history which I hope might help.
It's possible the Babylonians knew a version of Pythagoras' theorem back in 1700
BC—many centuries before Pythagoras—and likely that they did indeed treat it as "a fact of observation".
We know they could work out square roots, and there's a clay tablet with a great long list of Pythagorean triples on it.¹ They seem to have used quite a bit of maths, and they knew from experience that it worked. But there's no evidence of them trying to prove that it must work. They seem to have just used it.
What the Greeks did—people like Pythagoras, Euclid, and Archimedes—was set about understanding why things like Pythagoras' theorem always worked, and it's really with them that the idea of mathematical proofs began. Euclid's Elements is THE classic in this, and was used as a textbook in secondary schools right up to my father's time. Euclid put together a large amount of mathematics then known, and showed how it all held together.
Euclid started off with a few definitions that were considered so basic that nobody could disagree with them, and some statements that were so obvious and basic that they didn't need to be proved from anything else (and couldn't be anyway).²
The unprovable-but-obvious statements were (and are) called axioms. Euclid's included ones like
- Two things which are each equal to the same thing are equal to each other.
What he then did, step by step, was to show logically that if you accepted the definitions and axioms, you had no choice but to accept the various theorems he proved. A mathematical proof of a theorem is basically a demonstration that it inescapably follows either directly from the axioms, or from other things already proved from them.
A proof today doesn't usually start by spelling out the axioms it rests on. They're generally so basic and universally accepted that you'd only need to say if you weren't using one or other of them. But it will use various other theorems—maybe explicitly, maybe implied by using procedures that rely on them—and ultimately, if the proof is rock-solid, there's a chain we could follow all the way back to the axioms.
Notes
¹ Another tablet has a square drawn on it. Simplifying slightly, one side of the square is labelled $1$ and the diagonal is labelled with their equivalent of $1.41421$, ie an accurate value of $sqrt{2}$. (Actually it's written in base $60$, to $3$ base-60 significant figures, and expressed in $60$ths.)
² In fact, one of Euclid's axioms (the "parallel postulate") wasn't at all simple and obvious—but nobody could manage to prove it from the other axioms, and in the nineteenth century it was discovered that assuming it to be untrue leads to a whole new kind of geonetry, applicable to curved surfaces and spaces.
answered Dec 17 '18 at 19:17
timtfjtimtfj
2,468420
$endgroup$
add a comment |
$begingroup$
There are no mathematical truths in a metaphysical sense. What's exist is a consistent axiomatic method. Some "facts of observation" are selected as axioms and so called theorems - facts of observations or not - are proved to be consequences of the axioms by using inference rules as modus ponens and substitution.
answered Dec 22 '18 at 13:23
LehsLehs
7,02931664
$endgroup$
add a comment |
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2 Answers
2
active
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2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Here's some sketchy history which I hope might help.
It's possible the Babylonians knew a version of Pythagoras' theorem back in 1700
BC—many centuries before Pythagoras—and likely that they did indeed treat it as "a fact of observation".
We know they could work out square roots, and there's a clay tablet with a great long list of Pythagorean triples on it.¹ They seem to have used quite a bit of maths, and they knew from experience that it worked. But there's no evidence of them trying to prove that it must work. They seem to have just used it.
What the Greeks did—people like Pythagoras, Euclid, and Archimedes—was set about understanding why things like Pythagoras' theorem always worked, and it's really with them that the idea of mathematical proofs began. Euclid's Elements is THE classic in this, and was used as a textbook in secondary schools right up to my father's time. Euclid put together a large amount of mathematics then known, and showed how it all held together.
Euclid started off with a few definitions that were considered so basic that nobody could disagree with them, and some statements that were so obvious and basic that they didn't need to be proved from anything else (and couldn't be anyway).²
The unprovable-but-obvious statements were (and are) called axioms. Euclid's included ones like
- Two things which are each equal to the same thing are equal to each other.
What he then did, step by step, was to show logically that if you accepted the definitions and axioms, you had no choice but to accept the various theorems he proved. A mathematical proof of a theorem is basically a demonstration that it inescapably follows either directly from the axioms, or from other things already proved from them.
A proof today doesn't usually start by spelling out the axioms it rests on. They're generally so basic and universally accepted that you'd only need to say if you weren't using one or other of them. But it will use various other theorems—maybe explicitly, maybe implied by using procedures that rely on them—and ultimately, if the proof is rock-solid, there's a chain we could follow all the way back to the axioms.
Notes
¹ Another tablet has a square drawn on it. Simplifying slightly, one side of the square is labelled $1$ and the diagonal is labelled with their equivalent of $1.41421$, ie an accurate value of $sqrt{2}$. (Actually it's written in base $60$, to $3$ base-60 significant figures, and expressed in $60$ths.)
² In fact, one of Euclid's axioms (the "parallel postulate") wasn't at all simple and obvious—but nobody could manage to prove it from the other axioms, and in the nineteenth century it was discovered that assuming it to be untrue leads to a whole new kind of geonetry, applicable to curved surfaces and spaces.
answered Dec 17 '18 at 19:17
timtfjtimtfj
2,468420
$endgroup$
add a comment |
$begingroup$
Here's some sketchy history which I hope might help.
It's possible the Babylonians knew a version of Pythagoras' theorem back in 1700
BC—many centuries before Pythagoras—and likely that they did indeed treat it as "a fact of observation".
We know they could work out square roots, and there's a clay tablet with a great long list of Pythagorean triples on it.¹ They seem to have used quite a bit of maths, and they knew from experience that it worked. But there's no evidence of them trying to prove that it must work. They seem to have just used it.
What the Greeks did—people like Pythagoras, Euclid, and Archimedes—was set about understanding why things like Pythagoras' theorem always worked, and it's really with them that the idea of mathematical proofs began. Euclid's Elements is THE classic in this, and was used as a textbook in secondary schools right up to my father's time. Euclid put together a large amount of mathematics then known, and showed how it all held together.
Euclid started off with a few definitions that were considered so basic that nobody could disagree with them, and some statements that were so obvious and basic that they didn't need to be proved from anything else (and couldn't be anyway).²
The unprovable-but-obvious statements were (and are) called axioms. Euclid's included ones like
- Two things which are each equal to the same thing are equal to each other.
What he then did, step by step, was to show logically that if you accepted the definitions and axioms, you had no choice but to accept the various theorems he proved. A mathematical proof of a theorem is basically a demonstration that it inescapably follows either directly from the axioms, or from other things already proved from them.
A proof today doesn't usually start by spelling out the axioms it rests on. They're generally so basic and universally accepted that you'd only need to say if you weren't using one or other of them. But it will use various other theorems—maybe explicitly, maybe implied by using procedures that rely on them—and ultimately, if the proof is rock-solid, there's a chain we could follow all the way back to the axioms.
Notes
¹ Another tablet has a square drawn on it. Simplifying slightly, one side of the square is labelled $1$ and the diagonal is labelled with their equivalent of $1.41421$, ie an accurate value of $sqrt{2}$. (Actually it's written in base $60$, to $3$ base-60 significant figures, and expressed in $60$ths.)
² In fact, one of Euclid's axioms (the "parallel postulate") wasn't at all simple and obvious—but nobody could manage to prove it from the other axioms, and in the nineteenth century it was discovered that assuming it to be untrue leads to a whole new kind of geonetry, applicable to curved surfaces and spaces.
answered Dec 17 '18 at 19:17
timtfjtimtfj
2,468420
$endgroup$
add a comment |
$begingroup$
Here's some sketchy history which I hope might help.
It's possible the Babylonians knew a version of Pythagoras' theorem back in 1700
BC—many centuries before Pythagoras—and likely that they did indeed treat it as "a fact of observation".
We know they could work out square roots, and there's a clay tablet with a great long list of Pythagorean triples on it.¹ They seem to have used quite a bit of maths, and they knew from experience that it worked. But there's no evidence of them trying to prove that it must work. They seem to have just used it.
What the Greeks did—people like Pythagoras, Euclid, and Archimedes—was set about understanding why things like Pythagoras' theorem always worked, and it's really with them that the idea of mathematical proofs began. Euclid's Elements is THE classic in this, and was used as a textbook in secondary schools right up to my father's time. Euclid put together a large amount of mathematics then known, and showed how it all held together.
Euclid started off with a few definitions that were considered so basic that nobody could disagree with them, and some statements that were so obvious and basic that they didn't need to be proved from anything else (and couldn't be anyway).²
The unprovable-but-obvious statements were (and are) called axioms. Euclid's included ones like
- Two things which are each equal to the same thing are equal to each other.
What he then did, step by step, was to show logically that if you accepted the definitions and axioms, you had no choice but to accept the various theorems he proved. A mathematical proof of a theorem is basically a demonstration that it inescapably follows either directly from the axioms, or from other things already proved from them.
A proof today doesn't usually start by spelling out the axioms it rests on. They're generally so basic and universally accepted that you'd only need to say if you weren't using one or other of them. But it will use various other theorems—maybe explicitly, maybe implied by using procedures that rely on them—and ultimately, if the proof is rock-solid, there's a chain we could follow all the way back to the axioms.
Notes
¹ Another tablet has a square drawn on it. Simplifying slightly, one side of the square is labelled $1$ and the diagonal is labelled with their equivalent of $1.41421$, ie an accurate value of $sqrt{2}$. (Actually it's written in base $60$, to $3$ base-60 significant figures, and expressed in $60$ths.)
² In fact, one of Euclid's axioms (the "parallel postulate") wasn't at all simple and obvious—but nobody could manage to prove it from the other axioms, and in the nineteenth century it was discovered that assuming it to be untrue leads to a whole new kind of geonetry, applicable to curved surfaces and spaces.
answered Dec 17 '18 at 19:17
timtfjtimtfj
2,468420
$endgroup$
Here's some sketchy history which I hope might help.
It's possible the Babylonians knew a version of Pythagoras' theorem back in 1700
BC—many centuries before Pythagoras—and likely that they did indeed treat it as "a fact of observation".
We know they could work out square roots, and there's a clay tablet with a great long list of Pythagorean triples on it.¹ They seem to have used quite a bit of maths, and they knew from experience that it worked. But there's no evidence of them trying to prove that it must work. They seem to have just used it.
What the Greeks did—people like Pythagoras, Euclid, and Archimedes—was set about understanding why things like Pythagoras' theorem always worked, and it's really with them that the idea of mathematical proofs began. Euclid's Elements is THE classic in this, and was used as a textbook in secondary schools right up to my father's time. Euclid put together a large amount of mathematics then known, and showed how it all held together.
Euclid started off with a few definitions that were considered so basic that nobody could disagree with them, and some statements that were so obvious and basic that they didn't need to be proved from anything else (and couldn't be anyway).²
The unprovable-but-obvious statements were (and are) called axioms. Euclid's included ones like
- Two things which are each equal to the same thing are equal to each other.
What he then did, step by step, was to show logically that if you accepted the definitions and axioms, you had no choice but to accept the various theorems he proved. A mathematical proof of a theorem is basically a demonstration that it inescapably follows either directly from the axioms, or from other things already proved from them.
A proof today doesn't usually start by spelling out the axioms it rests on. They're generally so basic and universally accepted that you'd only need to say if you weren't using one or other of them. But it will use various other theorems—maybe explicitly, maybe implied by using procedures that rely on them—and ultimately, if the proof is rock-solid, there's a chain we could follow all the way back to the axioms.
Notes
¹ Another tablet has a square drawn on it. Simplifying slightly, one side of the square is labelled $1$ and the diagonal is labelled with their equivalent of $1.41421$, ie an accurate value of $sqrt{2}$. (Actually it's written in base $60$, to $3$ base-60 significant figures, and expressed in $60$ths.)
² In fact, one of Euclid's axioms (the "parallel postulate") wasn't at all simple and obvious—but nobody could manage to prove it from the other axioms, and in the nineteenth century it was discovered that assuming it to be untrue leads to a whole new kind of geonetry, applicable to curved surfaces and spaces.
answered Dec 17 '18 at 19:17
timtfjtimtfj
2,468420
edited Dec 17 '18 at 22:40
answered Dec 17 '18 at 19:17
timtfjtimtfj
2,468420
answered Dec 17 '18 at 19:17
timtfjtimtfj
2,468420
answered Dec 17 '18 at 19:17
timtfjtimtfj
2,468420
2,468420
add a comment |
add a comment |
$begingroup$
There are no mathematical truths in a metaphysical sense. What's exist is a consistent axiomatic method. Some "facts of observation" are selected as axioms and so called theorems - facts of observations or not - are proved to be consequences of the axioms by using inference rules as modus ponens and substitution.
answered Dec 22 '18 at 13:23
LehsLehs
7,02931664
$endgroup$
add a comment |
$begingroup$
There are no mathematical truths in a metaphysical sense. What's exist is a consistent axiomatic method. Some "facts of observation" are selected as axioms and so called theorems - facts of observations or not - are proved to be consequences of the axioms by using inference rules as modus ponens and substitution.
answered Dec 22 '18 at 13:23
LehsLehs
7,02931664
$endgroup$
add a comment |
$begingroup$
There are no mathematical truths in a metaphysical sense. What's exist is a consistent axiomatic method. Some "facts of observation" are selected as axioms and so called theorems - facts of observations or not - are proved to be consequences of the axioms by using inference rules as modus ponens and substitution.
answered Dec 22 '18 at 13:23
LehsLehs
7,02931664
$endgroup$
There are no mathematical truths in a metaphysical sense. What's exist is a consistent axiomatic method. Some "facts of observation" are selected as axioms and so called theorems - facts of observations or not - are proved to be consequences of the axioms by using inference rules as modus ponens and substitution.
answered Dec 22 '18 at 13:23
LehsLehs
7,02931664
edited Dec 22 '18 at 13:42
answered Dec 22 '18 at 13:23
LehsLehs
7,02931664
answered Dec 22 '18 at 13:23
LehsLehs
7,02931664
answered Dec 22 '18 at 13:23
LehsLehs
7,02931664
7,02931664
add a comment |
add a comment |
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"What exactly are you proving?" Mathematical statements, often written in symbolic form.
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– Mauro ALLEGRANZA
Dec 17 '18 at 16:26
1
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Many mathematical statements, like e.g. arithmetical and geometrical ones are used in "everyday life" : to count colelctions of objects, to measure physical and "practical" magnitudes.
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– Mauro ALLEGRANZA
Dec 17 '18 at 16:28
1
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A proof is the derivation of a statement from specific axioms (i.e. the axioms assumed as basic facts of the discipline involving the said statement) using rules of logic (and previously proved statements, called theorems).
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– Mauro ALLEGRANZA
Dec 17 '18 at 16:29
3
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Not at all; the Pythagorean theorem has been proved e.g. by Euclid (ca.300 BCE, Ancient Greece) for geometrical axioms (see Euclid's Elements).
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– Mauro ALLEGRANZA
Dec 17 '18 at 16:30
2
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It might help to look at an example of a mathematical proof before asking a question like this. If you google, for example, "proof of the Pythagorean theorem" you'll see some indication of what a mathematical proof consists of. It's not simply "we always see that it works."
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– Noah Schweber
Dec 17 '18 at 16:33