Why mathematics is neither absolutely nor objectively "right."

[lostone]

Calling it a successor is a self defined rule. Very much like asserting that parallel lines can never meet.

 

[Unknown Soldier]

Calling it a successor is a self defined rule.

How can a rule define itself? But I think you mean that the successor rule is arbitrarily defined which it is, of course. It does seem to be an intuitive rule though because every time you add one of something, the total usually increases.

Very much like asserting that parallel lines can never meet.

Yes. If every point on a line is always at least a fixed distance from any point on another line on the same plane, then those lines are defined to be parallel. It's an intuitive idea because we can draw lines on a plane that are always that distance from each other. So although math is based on arbitrary rules, those rules most often reflect "the real world" and therefore seem reasonable.

So it's great to see that I'm finally getting through on this issue.

 

[pood]

Calling it a successor is a self defined rule.

How can a rule define itself? But I think you mean that the successor rule is arbitrarily defined which it is, of course. It does seem to be an intuitive rule though because every time you add one of something, the total usually increases.

Very much like asserting that parallel lines can never meet.

Yes. If every point on a line is always at least a fixed distance from any point on another line on the same plane, then those lines are defined to be parallel. It's an intuitive idea because we can draw lines on a plane that are always that distance from each other. So although math is based on arbitrary rules, those rules most often reflect "the real world" and therefore seem reasonable.

So it's great to see that I'm finally getting through on this issue.

I wouldn’t flatter yourself that you are “getting through” to people. I doubt anyone here disagrees with this, or with what Singham says, but what you are saying now seems to be a significant backtrack from what you were saying, or at least seeming to imply, initially.

Singham makes a distinction between maths that are “right” and “true,” and something like Euclidian geometry has always been “right,” even self-evidently so. Until relatively recently it was also throught to be “true,” defined here as accurately mathematically describing the real world. But general relatively showed it was not “true” in that sense, and it had to be replaced by a geometry describing geodesics.

The maths of string theory are certainly “right,” but whether they accurately describe the real world is unknown. Newton’s equations get us to the planets just fine but break down at relativistic velocities and in the quantum micro world. Yet relativistic and quantum maths must also break down at some level of description because GR and quantum theory are in conflict.

As described in the link earlier, if I have a clock that counts only to twelve but must describe 24 separate numbers, we get a situation in which the number 17 ultimately translates to the number five, but the key word is translates. The translation maintains the same, objective value. We use base 10 arithmetic probably because we have ten fingers, but we can use any base we want and get seemingly different values, yet these values will all seamlessly translate into one another to maintain the objectivity of math. In the sense of being “right,” math is indeed objective and ultimately probably just a set of increasingly complicated tautologies. But the question of whether our mathematical structures are not just right but true in the sense of whether they describe what’s in the real world is an entirely different question and not the issue that you were raising initially, in so far as I can tell.

 

[Unknown Soldier]

I wouldn’t flatter yourself that you are “getting through” to people.

So right off the bat you take a swat at me. What about the topic? Isn't that important to you?

I doubt anyone here disagrees with this, or with what Singham says, but what you are saying now seems to be a significant backtrack from what you were saying, or at least seeming to imply, initially.

You evidently haven't bothered to read much of the thread. Again and again it has been asserted that math is absolutely and objectively right. That sounds like disagreement to me.

Singham makes a distinction between maths that are “right” and “true,” and something like Euclidian geometry has always been “right,” even self-evidently so.

You're making one of the biggest goofs in mathematics: You are relying on intuition. Mathematicians and philosophers of mathematics know full well the pitfalls of relying on intuition because intuition is often wrong. That's why mathematicians only conclude that some conjecture is true if it is rigorously proved using valid logic.

Until relatively recently it was also throught to be “true,” defined here as accurately mathematically describing the real world. But general relatively showed it was not “true” in that sense, and it had to be replaced by a geometry describing geodesics.

You're moving out of the realm of mathematics to physics here. The topic is if mathematics true, not if some scientific theory is true.

As described in the link earlier, if I have a clock that counts only to twelve but must describe 24 separate numbers, we get a situation in which the number 17 ultimately translates to the number five, but the key word is translates. The translation maintains the same, objective value. We use base 10 arithmetic probably because we have ten fingers, but we can use any base we want and get seemingly different values, yet these values will all seamlessly translate into one another to maintain the objectivity of math. In the sense of being “right,” math is indeed objective and ultimately probably just a set of increasingly complicated tautologies. But the question of whether our mathematical structures are not just right but true in the sense of whether they describe what’s in the real world is an entirely different question and not the issue that you were raising initially, in so far as I can tell.

But those "translations" you mention here are arbitrary. In other words, the symbol "2" can be given different meanings simply by changing the definition of what it means and the rules of operations on it. It then has no "objective value," of course.

Finally, if math is objective, then I'm wondering why we're disagreeing about it. But I do understand that "objective" is a very strong and impressive adjective. It connotes stability and reliability. And who wants to live in a world where everything including math is just another idea somebody made up?

 

[pood]

I wouldn’t flatter yourself that you are “getting through” to people.

So right off the bat you take a swat at me. What about the topic? Isn't that important to you?

I’m pointing out that it’s self-serving and more than a little condescending to claim you are making headway with others when it seems you are not. If others agree with you and not with me on this matter, they can speak for themselves.

I doubt anyone here disagrees with this, or with what Singham says, but what you are saying now seems to be a significant backtrack from what you were saying, or at least seeming to imply, initially.

You evidently haven't bothered to read much of the thread. Again and again it has been asserted that math is absolutely and objectively right. That sounds like disagreement to me.

But you claim you agree with Singham and he is saying that math IS objectively right, within its own starting assumptions. So if you agree with him, you must now be disagreeing with yourself. In any case, he’s making a useful distinction between whether it’s right and whether it’s TRUE, i.e. whether it describes the real world.

Singham makes a distinction between maths that are “right” and “true,” and something like Euclidian geometry has always been “right,” even self-evidently so.

You're making one of the biggest goofs in mathematics: You are relying on intuition. Mathematicians and philosophers of mathematics know full well the pitfalls of relying on intuition because intuition is often wrong. That's why mathematicians only conclude that some conjecture is true if it is rigorously proved using valid logic.

I’m afraid this is completely wrong and not at all what I said or implied. The postulates of Euclidean geometry are absolutely and objectively correct, assuming the truth of the starting axioms. Now whether the axioms and the theorems that follow from are TRUE or not (describe the real world) is a different matter, as I pointed out. It turns out they aren’t true. I’ve made no “goof” at all.

Until relatively recently it was also throught to be “true,” defined here as accurately mathematically describing the real world. But general relatively showed it was not “true” in that sense, and it had to be replaced by a geometry describing geodesics.

You're moving out of the realm of mathematics to physics here. The topic is if mathematics true, not if some scientific theory is true.

And yet you yourself endorsed the writing of Singham who is doing that very thing.

As described in the link earlier, if I have a clock that counts only to twelve but must describe 24 separate numbers, we get a situation in which the number 17 ultimately translates to the number five, but the key word is translates. The translation maintains the same, objective value. We use base 10 arithmetic probably because we have ten fingers, but we can use any base we want and get seemingly different values, yet these values will all seamlessly translate into one another to maintain the objectivity of math. In the sense of being “right,” math is indeed objective and ultimately probably just a set of increasingly complicated tautologies. But the question of whether our mathematical structures are not just right but true in the sense of whether they describe what’s in the real world is an entirely different question and not the issue that you were raising initially, in so far as I can tell.

But those "translations" you mention here are arbitrary. In other words, the symbol "2" can be given different meanings simply by changing the definition of what it means and the rules of operations on it. It then has no "objective value," of course.

The translations are not arbitrary. Are you suggesting that translations between different bases in arithmetic are arbitrary?

To return to the clock example: Are you saying the translation from 17 to 5 is arbitrary? I say it is not. The translation follows as a logical consequence of trying to fit 24 numbers into a counting system that ends with the number 12.

Finally, if math is objective, then I'm wondering why we're disagreeing about it. But I do understand that "objective" is a very strong and impressive adjective. It connotes stability and reliability. And who wants to live in a world where everything including math is just another idea somebody made up?

Math isn’t an idea that somebody ”made up.”

 

[steve_bank]

Yes. If every point on a line is always at least a fixed distance from any point on another line on the same plane, then those lines are defined to be parallel. It's an intuitive idea because we can draw lines on a plane that are always that distance from each other. So although math is based on arbitrary rules, those rules most often reflect "the real world" and therefore seem reasonable.

So it's great to see that I'm finally getting through on this issue.

Hmmm... So, lines ar only prlell in aplabe? What about two lines in three dinesions that are not on a plane?


For two lines in an xy plane, matematecally how would you prove the lines are patarallel?

Hint: y = mx + b.

To philosophize about math is human, to actually do math is divine.

Genaeraly speaking science is considered objective because it deals in physically measurable and uantifiable variables not subjct to interpretion. That an object weighs 1kg does not change based on how you philiosphcaly view science.

US would rersort to sophistry and ague the definition of a kg is arbireay which it is and then conclude it s subkective. The argument becomes semantics as does a lot of philosophy.

There have been several lengthy threads on objective vs subjective evidence.

To me objective applied to science and nath means not subject to personal interpretative as to results of a thematically operation.

That 1 rock plus 1 rock means there are 2 rocks on total is objective observation.

Arbitarry does not men subjective.

If I define the bank as the distance between my outstretched finger tips and say the distance bewteen two points is 10 banks, that is objective observation and quantification.


The meter, kilogram, and second and SI units are indeed arbitry, but are objective bot subjective.

Math and scince are products of te human brain so in a philosphcal snse are imagined. That does not mean math as applied is subjective.


At the end of an NBA game the team which has the highest numbner of counted baskets wins. Nothing subjective to it.

 

[steve_bank]

US

1 + 2 * 3

A;algebraically is the correct answer 7 0r 9 and why? Can it be subjectively interpreted as anything but one correct answer?

Same with 6 / 2 + 1. Is it 4 or 2? Can it be interpreted in two ways?

You have the right to remain silent. Anything you say can and will be used against you.

 

[Unknown Soldier]

I wouldn’t flatter yourself that you are “getting through” to people.

So right off the bat you take a swat at me. What about the topic? Isn't that important to you?

I’m pointing out that it’s self-serving and more than a little condescending to claim you are making headway with others when it seems you are not. If others agree with you and not with me on this matter, they can speak for themselves.

I'm getting people to understand or agree to what I posted in the OP. That's a simple fact. How condescending you find me stating that fact has nothing to do with the topic. So let's lay off the ad hominem arguments. Agreed?

You evidently haven't bothered to read much of the thread. Again and again it has been asserted that math is absolutely and objectively right. That sounds like disagreement to me.

But you claim you agree with Singham and he is saying that math IS objectively right, within its own starting assumptions. So if you agree with him, you must now be disagreeing with yourself. In any case, he’s making a useful distinction between whether it’s right and whether it’s TRUE, i.e. whether it describes the real world.

It's best to post direct quotations from a source. Here is what Singham actually said:

Even a statement such as “1+1=2,” which most people might regard as a universal truth that cannot be denied, is seen by them [pure mathematicians] as merely the consequence of certain starting assumptions, and one cannot assign any absolute truth value to it.

Nowhere does he say that "math is objectively right within its own starting assumptions." You misquoted him. If he did say what you claim, then he would be contradicting himself because something that is objectively right cannot be based on assumptions! He like I recognizes that math involves no undeniable universal truths--exactly what I've argued from the OP. And all that in the face of repeated denials on the part of numerous other members here.

Singham makes a distinction between maths that are “right” and “true,” and something like Euclidian geometry has always been “right,” even self-evidently so.

You're making one of the biggest goofs in mathematics: You are relying on intuition. Mathematicians and philosophers of mathematics know full well the pitfalls of relying on intuition because intuition is often wrong. That's why mathematicians only conclude that some conjecture is true if it is rigorously proved using valid logic.

I’m afraid this is completely wrong and not at all what I said or implied. The postulates of Euclidean geometry are absolutely and objectively correct, assuming the truth of the starting axioms. Now whether the axioms and the theorems that follow from are TRUE or not (describe the real world) is a different matter, as I pointed out. It turns out they aren’t true. I’ve made no “goof” at all.

OK, yes sometimes unprovable postulates like those in Euclidian geometry are unavoidable. But contrary to what you say that doesn't make them "absolutely and objectively correct." Again, objective truth cannot be based on arbitrary rules like axioms or on assumptions. If objective truth could be established that way, then the rules of baseball would be absolute truths.

You're moving out of the realm of mathematics to physics here. The topic is if mathematics true, not if some scientific theory is true.

And yet you yourself endorsed the writing of Singham who is doing that very thing.

Again, let's look at what Singham really said rather than rely on your paraphrase:

… So while in mathematics the statement “1+1-2” is simply a string of symbols representing a theorem based on a particular set of axioms and rules of logic, in science, its empirical truth or falsity is extremely important and is judged by how well real objects (apples, chairs, etc.) conform to it.

Singham here is merely explaining that math should be useful in real-world applications and not that it's important to know if some scientific theory is true.

But those "translations" you mention here are arbitrary. In other words, the symbol "2" can be given different meanings simply by changing the definition of what it means and the rules of operations on it. It then has no "objective value," of course.

The translations are not arbitrary. Are you suggesting that translations between different bases in arithmetic are arbitrary?

In that nothing in math is absolute, yes they are arbitrary. Number bases are based on rules made up by mathematicians.

To return to the clock example: Are you saying the translation from 17 to 5 is arbitrary? I say it is not. The translation follows as a logical consequence of trying to fit 24 numbers into a counting system that ends with the number 12.

Change the rules, and you change what "17" might be. I can make up a rule that 17 is a cow and 18 is a bull. You don't see that as arbitrary?

Finally, if math is objective, then I'm wondering why we're disagreeing about it. But I do understand that "objective" is a very strong and impressive adjective. It connotes stability and reliability. And who wants to live in a world where everything including math is just another idea somebody made up?

Math isn’t an idea that somebody ”made up.”

That's right. Math is a set of ideas that people have made up.

 

[pood]

I wouldn’t flatter yourself that you are “getting through” to people.

So right off the bat you take a swat at me. What about the topic? Isn't that important to you?

I’m pointing out that it’s self-serving and more than a little condescending to claim you are making headway with others when it seems you are not. If others agree with you and not with me on this matter, they can speak for themselves.

I'm getting people to understand or agree to what I posted in the OP. That's a simple fact.

Is it?

How condescending you find me stating that fact has nothing to do with the topic. So let's lay off the ad hominem arguments. Agreed?

I didn’t ad hom you. Not even close.

You evidently haven't bothered to read much of the thread. Again and again it has been asserted that math is absolutely and objectively right. That sounds like disagreement to me.

But you claim you agree with Singham and he is saying that math IS objectively right, within its own starting assumptions. So if you agree with him, you must now be disagreeing with yourself. In any case, he’s making a useful distinction between whether it’s right and whether it’s TRUE, i.e. whether it describes the real world.

It's best to post direct quotations from a source. Here is what Singham actually said:

Even a statement such as “1+1=2,” which most people might regard as a universal truth that cannot be denied, is seen by them [pure mathematicians] as merely the consequence of certain starting assumptions, and one cannot assign any absolute truth value to it.

Nowhere does he say that "math is objectively right within its own starting assumptions." You misquoted him. If he did say what you claim, then he would be contradicting himself because something that is objectively right cannot be based on assumptions! He like I recognizes that math involves no undeniable universal truths--exactly what I've argued from the OP. And all that in the face of repeated denials on the part of numerous other members here.

But Singham goes on to state exactly the same thing I said with respect to Euclidean geometry. Go back and read what he wrote. As I noted, you start with certain axioms that are assumed to be true. Whether they actually are true is an empirical matter. Euclidean geometry fails the empirical test but 1+1=2 passes. As I argued earlier, in a world where a third object invariably appears whenever two objects are brought together, it seems we would have to say that 1+1=3.

Singham makes a distinction between maths that are “right” and “true,” and something like Euclidian geometry has always been “right,” even self-evidently so.

You're making one of the biggest goofs in mathematics: You are relying on intuition. Mathematicians and philosophers of mathematics know full well the pitfalls of relying on intuition because intuition is often wrong. That's why mathematicians only conclude that some conjecture is true if it is rigorously proved using valid logic.

I’m afraid this is completely wrong and not at all what I said or implied. The postulates of Euclidean geometry are absolutely and objectively correct, assuming the truth of the starting axioms. Now whether the axioms and the theorems that follow from are TRUE or not (describe the real world) is a different matter, as I pointed out. It turns out they aren’t true. I’ve made no “goof” at all.

OK, yes sometimes unprovable postulates like those in Euclidian geometry are unavoidable. But contrary to what you say that doesn't make them "absolutely and objectively correct." Again, objective truth cannot be based on arbitrary rules like axioms or on assumptions. If objective truth could be established that way, then the rules of baseball would be absolute truths.

Of course objective rightness can be bsed on arbitary rules, such as Euclid’s starting axioms. The theorems follow as logical truths derived from the axioms. But Euclidean geometry is not true in reality. I think this is the distinction Singham makes between ”right” and “true.”

You're moving out of the realm of mathematics to physics here. The topic is if mathematics true, not if some scientific theory is true.

And yet you yourself endorsed the writing of Singham who is doing that very thing.

Again, let's look at what Singham really said rather than rely on your paraphrase:

… So while in mathematics the statement “1+1-2” is simply a string of symbols representing a theorem based on a particular set of axioms and rules of logic, in science, its empirical truth or falsity is extremely important and is judged by how well real objects (apples, chairs, etc.) conform to it.

Singham here is merely explaining that math should be useful in real-world applications and not that it's important to know if some scientific theory is true.

I didn’t say it’s important to know if some scientific theory is true. I said exactly what he said. Reread what I wrote, particularly my post on how Newtonian mechanics, relativity and quantum maths relate to reality.

But those "translations" you mention here are arbitrary. In other words, the symbol "2" can be given different meanings simply by changing the definition of what it means and the rules of operations on it. It then has no "objective value," of course.

The translations are not arbitrary. Are you suggesting that translations between different bases in arithmetic are arbitrary?

In that nothing in math is absolute, yes they are arbitrary. Number bases are based on rules made up by mathematicians.

Of course they are, but so what?

To return to the clock example: Are you saying the translation from 17 to 5 is arbitrary? I say it is not. The translation follows as a logical consequence of trying to fit 24 numbers into a counting system that ends with the number 12.

Change the rules, and you change what "17" might be. I can make up a rule that 17 is a cow and 18 is a bull. You don't see that as arbitrary?

Yes, you could do that, except that it would have nothing to do with arithmetic or mathematics. In the clock example, 17 = 5 as a logical consequence of the translation of 24 hours counted on a clock with 12 numbers. It cannot be otherwise. You may invent different labels for the numbers, but their values are identical.

Finally, if math is objective, then I'm wondering why we're disagreeing about it. But I do understand that "objective" is a very strong and impressive adjective. It connotes stability and reliability. And who wants to live in a world where everything including math is just another idea somebody made up?

Math isn’t an idea that somebody ”made up.”

That's right. Math is a set of ideas that people have made up.

Math is invented, not made up. There is a difference.

 

[steve_bank]

Math is invented. not exact;y breaking news.

Words are invented, like the owed absolute. The word absolute is contextual and, contxtual, and emotionaly loaded.

Is the meaning of the word absolute absolute in itself?

 

[steve_bank]

US

1 + 2 * 3

A;algebraically is the correct answer 7 0r 9 and why? Can it be subjectively interpreted as anything but one correct answer?

Same with 6 / 2 + 1. Is it 4 or 2? Can it be interpreted in two ways?

You have the right to remain silent. Anything you say can and will be used against you.

Apparently US skipped high school math.

 

[Eldarion Lathria]

Everyone needs to remember the 'academic and sociological' answer to the the question "What is 2 + 2?"

Eldarion Lathria

 

[Unknown Soldier]

I'm getting people to understand or agree to what I posted in the OP. That's a simple fact.

Is it?

Yes. Unfortunately there was a lot of trolling on this thread and there still is, but at least some people are honest enough now to accept that the OP is right.

How condescending you find me stating that fact has nothing to do with the topic. So let's lay off the ad hominem arguments. Agreed?

I didn’t ad hom you. Not even close.

Just don't do it.

But Singham goes on to state exactly the same thing I said with respect to Euclidean geometry. Go back and read what he wrote. As I noted, you start with certain axioms that are assumed to be true. Whether they actually are true is an empirical matter. Euclidean geometry fails the empirical test but 1+1=2 passes. As I argued earlier, in a world where a third object invariably appears whenever two objects are brought together, it seems we would have to say that 1+1=3.

Actually, what we say in mathematics is often independent of empirical evidence although much of what is asserted in mathematics is based on "the real world." I think that's why many people are so amazed when they see the consistency between math and the cosmos thinking that the cosmos conforms to math. Of course, it's the other way around.

Of course objective rightness can be bsed on arbitary rules, such as Euclid’s starting axioms. The theorems follow as logical truths derived from the axioms. But Euclidean geometry is not true in reality. I think this is the distinction Singham makes between ”right” and “true.”

No. I'm afraid not. You can't create truth for everybody based on what you made up yourself.

I didn’t say it’s important to know if some scientific theory is true. I said exactly what he said. Reread what I wrote, particularly my post on how Newtonian mechanics, relativity and quantum maths relate to reality.

No. If it was wrong before, then it won't be right now.

In that nothing in math is absolute, yes they are arbitrary. Number bases are based on rules made up by mathematicians.

Of course they are, but so what?

So I'm right about what I've been arguing since the OP.

Change the rules, and you change what "17" might be. I can make up a rule that 17 is a cow and 18 is a bull. You don't see that as arbitrary?

Yes, you could do that, except that it would have nothing to do with arithmetic or mathematics.

Uh boy--OK, we're talking about definitions in mathematics. OK? So if a definition about a number can be arbitrarily assigned "cow" or "bull," then that demonstrates that all definitions are arbitrary including the definitions in mathematics.

In the clock example, 17 = 5 as a logical consequence of the translation of 24 hours counted on a clock with 12 numbers. It cannot be otherwise. You may invent different labels for the numbers, but their values are identical.

Sure. And if you change the rules, then 17 will equal something else. It is only true that "17 = 5" within the set of rules you make up. It appears that you are confusing consistency with what is absolute. 17 = 5 may be consistent with the rules you are using, but it isn't absolute, of course.

That's right. Math is a set of ideas that people have made up.

Math is invented, not made up. There is a difference.

Hmmm. Not in my dictionary. But are you referencing an absolutely right dictionary?