Maths works in the sense that it doesn’t generate contradictions,and if it didn’t work that way, it would be no good as a necessary ingredient of physics.
Right. So the fact that it works as a necessary ingredient of physics is evidence of its general ability to derive true statements from other true statements.
But a necessary ingredient is not a sufficient ingredient. You don’t always get the same result from maths and physics, as the example of apples, inkblots and rabbits shows.
Combining ink blobs and mating rabbits are not properly represented by the mathematical operation of addition, that’s simply an example of using the wrong math.
You asked me, several comments upthread, whether I thought the universe would be observably different if math didn’t work. The fact that not all possible mathematical statements reflect reality doesn’t get us around the answer being “yes.”
So the fact that [math] works as a necessary ingredient of physics is evidence of its general ability to derive true statements from other true statements.
Which does not itself support your claim that it has the same epistemology as physical science.
Combining ink blobs and mating rabbits are not properly represented by the mathematical operation of addition, that’s simply an example of using the wrong math.
But it’s not seen as a disproof of any mathematical claim....although it would be a disproof of a physical theory, if it mispredicted an outcome … because physics has an empirical epistemology and maths doesn’t.
You asked me, several comments upthread, whether I thought the universe would be observably different if math didn’t work.
Which was part of a wider point about whether maths has an empirical epistemology, which was part of a wider point about moral claims having a sui generis epistemology.
The fact that not all possible mathematical statements reflect reality doesn’t get us around the answer being “yes.”
You expect reality to be different if deriving-a-true-statement-from-another-true-statement is different? But it is different. You can add or drop the Axiom of Choice. You can add or drop proof by Reductio (constructivism). Etc.
Which does not itself support your claim that it has the same epistemology as physical science.
This is not a claim I ever made in the first place. The fact that we do not use empirical investigation to determine the truth of individual mathematical claims is irrelevant to my point.
There’s nothing wrong with moral data being generated by a different sort of process than we use to generate scientific data, if you can demonstrate that the process works in the first place.
Absent that kind of evidence, evidence we have in overabundance for math, you should make no such assumption.
There’s nothing wrong with moral data being generated by a different sort of process than we use to generate scientific data, if you can demonstrate that the process works in the first place.
Do we actually have any such proofs? You seem to think maths can be proven to work empoirically, but that seems to depend on cherry picking the right maths and the right physical objects.
You seem to think maths can be proven to work empoirically, but that seems to depend on cherry picking the right maths and the right physical objects.
Can you give examples of math not working then? The fact that you can combine two blobs of ink to get one blob of ink has no bearing on the correctness of 1+1=2, because “+” is not an appropriate operator to represent mushing two objects together until they merge and then counting the merged object.
The fact that you get the wrong answers to your questions if you attempt to answer them with the wrong math is not relevant to the proposition that math works, just as the fact that using the wrong map will not help you navigate to your destination is irrelevant to the proposition that cartography works.
Do we really have an abundance? As far as I know, the non-circular vindication of popular forms of epistemollogical justification is a complex unsolved problem.
I can give examples of examples of correct maths that are also examples of incorrect maths. I can easily give examples of correct maths that are incorrect physics. “Working” has no single meaning here.
1+1=2 is a mathematical truth, period. If it doesn’t apply to inkdrops, that is a physics issue.
I amnot attempting to argue that maths does not work. I am arguing that it does not work like physics. It has a different epistemology.
It’s not that the equation doesn’t apply to inkdrops, it’s just that it doesn’t apply to doing that particular operation on inkdrops. 1+1=2 applies to inkdrops, as it does to any other physical objects, as long as you put them alongside each other and count them separately, rather than mashing them together and counting them as a unit. The operation of 1+1=2 never applies to mashing individual things together and counting how many of the mashed-together unit you have, although other qualities such as their masses will continue to be additive.
I am not arguing that mathematical truths are generated by the same epistemology as scientific data, but I am arguing that the entire edifice of math is supported by evidence, in the same manner that the edifice of the scientific method is supported by evidence, and that we do not likewise have evidence to support the edifice of an epistemology for generating objective moral truths.
I am arguing that the entire edifice of math is supported by evidence, in the same manner that the edifice of the scientific method is supported by evidence, and that we do not likewise have evidence to support the edifice of an epistemology for generating objective moral truths.
You can go upwards in the tree of comments to see where I’ve discussed this already, but if you still find my meaning unclear I could answer whatever questions you might have to clarify my position.
OK, thanks. I have looked through the thread again, and it looks like our views are too far apart for a forum exchange to be an adequate medium of discussing them. Or maybe I need to express them to myself clearer first.
It’s not that the equation doesn’t apply to inkdrops, it’s just that it doesn’t apply to doing that particular operation on inkdrops.
That’s a difference that doesn’t make a difference. It remains the case that a mathematical truth is not automatically a physical truth.
1+1=2 applies to inkdrops, as it does to any other physical objects, as long as you put them alongside each other and count them separately, rather than mashing them together and counting them as a unit. The operation of 1+1=2 never applies to mashing individual things together and counting how many of the mashed-together unit you have, although other qualities such as their masses will continue to be additive.
It remains the case that the skill, activity, and concern of picking out the bits of maths that are relevant to a given physical situation is physics
I am not arguing that mathematical truths are generated by the same epistemology as scientific data, but I am arguing that the entire edifice of math is supported by evidence, in the same manner that the edifice of the scientific method is supported by evidence
Then what’s the difference?
And what difference is evidence making to maths? You have axioms, which are true by stipulation, and you have truth -preserving rules of inference. Logically, if you combine the two, you will generate true theorems. So where
is the problem? Is the evidence supposed to remove doubt that the rules of inference are valid, or that the axioms are true?
And what difference is evidence making to maths? You have axioms, which are true by stipulation, and you have truth -preserving rules of inference. Logically, if you combine the two, you will generate true theorems. So where is the problem? Is the evidence supposed to remove doubt that the rules of inference are valid, or that the axioms are true?
The former. Some axioms will be applicable to reality and some will not, but the rules of inference work in either case.
Scientists tend to arrive at consensuses because they’re working with the same fundamental rules of inference and looking at the same data. Mathematicians arrive at consensuses on the consequences of given axioms, because they’re working with the same fundamental rules of inference. In both cases, we have evidence that these rules of inference actually work, which is not the case by default.
Even if we supposed that we had a system for moral reasoning which, like math, could extrapolate the consequences of axioms to their necessary conclusions (which I would argue we don’t actually have, humans are simply not good enough at natural language reasoning to pull that off with any sort of reliability,) how would we determine if any of those moral axioms apply to the real world, the way some mathematical axioms observably do?
The former. Some axioms will be applicable to reality and some will not, but the rules of inference work in either case.
And if some axioms are applicable to reality, then the whole edifice is supported? You mean, maths is like a rigid structure, where if one part supported, that in turn supports the other parts? That would be the case if you could logically infer axioms B and C from axiom A. But that is exactly how axioms don’t work. A set of axioms is a minimal set of assumptions: if one of your axioms is derivable from the others, it shouldn’t be in the set.
Maths doesn’t need empirical support to work as maths, since it isn’t physics, and adding empirical support
doesn’t show “it works” in any exhaustive way, only that some bits of it are applicable to physical reality,
In both cases, we have evidence that these rules of inference actually work, which is not the case by default.
Whatever “actually works” means. If a true mathematical statement is one that is derived by standard rules of inference from standard axioms, then it could hardly fail to “work”, in that sense. OTOH, it can still fail to apply
to reality, and so fail to work in that sense. (You really need to taboo “work”). But in the second case, evidence is only supporting the idea that some maths works, in the second sense. You make things rather easy by defining works to mean sometimes works...
Even if we supposed that we had a system for moral reasoning which, like math, could extrapolate the consequences of axioms to their necessary conclusions (which I would argue we don’t actually have, humans are simply not good enough at natural language reasoning to pull that off with any sort of reliability,) how would we determine if any of those moral axioms apply to the real world, the way some mathematical axioms observably do?
The job of morality is to guide action, not to passively reflect facts, so the question is irrelevant.
Whatever “actually works” means. If a true mathematical statement is one that is derived by standard rules of inference from standard axioms, then it could hardly fail to “work”, in that sense.
The “standard rules of inference” could very easily simply be nonsense. Sets of rules do not help produce meaningful information by default. Out of the set of all methods of inference that could hypothetically exist, the vast majority do not produce information that is either true or consistent.
The job of morality is to guide action, not to passively reflect facts, so the question is irrelevant.
So let’s suppose that we live in a universe in which it is objectively true that there are no objective moral standards. In your view, would this be a relevant fact about morality?
The “standard rules of inference” could very easily simply be nonsense.
In what sense? Are there alternative rules that are also truth preserving? Is truth-preservation hard to establish?
Sets of rules do not help produce meaningful information by default.
I wouldn’t say they produceinformation at all. Truth isn’t meaning, preservation isn’t production, truth preservation therefore isn’t meaning production.
Out of the set of all methods of inference that could hypothetically exist, the vast majority do not produce information that is either true or consistent.
We are supposed to have already chosen a set of rules that is truth-preserving.: ie, if you treat truth as a sort of numerical value, wihout any metaphysics behind it, that is assigned by fiat to your axioms, then you can show that it is preserved by using truth tables and the like (although there as some circularities there). Do you doubt that? Or is your concern showing that axioms are somehow “really” true, about something other than themselves?
So let’s suppose that we live in a universe in which it is objectively true that there are no objective moral standards. In your view, would this be a relevant fact about morality?
It’s more metaethical. You cant use “there are not moral truths” to guide your actions at the object level..
In what sense? Are there alternative rules that are also truth preserving? Is truth-preservation hard to establish?
If you count empiricism as a set of alternative rules which are truth preserving, then yes. If you’re talking about other non-empirical sets of rules, I would at least tentatively say no; formal logic is a branch of math, and natural language logic really doesn’t meet the same standards.
I wouldn’t say they produceinformation at all. Truth isn’t meaning, preservation isn’t production, truth preservation therefore isn’t meaning production.
In that case, what if anything would you describe as the production of information?
We are supposed to have already chosen a set of rules that is truth-preserving.: ie, if you treat truth as a sort of numerical value, wihout any metaphysics behind it, that is assigned by fiat to your axioms, then you can show that it is preserved by using truth tables and the like (although there as some circularities there). Do you doubt that? Or is your concern showing that axioms are somehow “really” true, about something other than themselves?
What set of rules “we are supposed to have already chosen” are you referring to?
Based on some of your statements about morality so far, I suspect I would argue that you are extrapolating from axioms that are not true in real life, much as some mathematical axioms seem to apply to the real world, while other ones do not.
It’s more metaethical. You cant use “there are not moral truths” to guide your actions at the object level..
Sure you can. Whether you should is an entirely separate value judgment.
If empiricism just means gathering observations, stamp collecting style, then i dont see where the truth preservation comes in. I can see where truth preservation comes into making predictions from evidence you already have, but that seems to use logical inference.
If you draw a line around an entity, then there will be information crossing that line, which is new, unpredictable, and “produced” as far as that entity is concerned.
I meant we are supposed to have already chosen a set of truth preserving mathematical and logical rules.
Could you give an example of how to use “there are no moral truths” to guide your actions?
If empiricism just means gathering observations, stamp collecting style, then i dont see where the truth preservation comes in. I can see where truth preservation comes into making predictions from evidence you already have, but that seems to use logical inference.
Could you taboo “truth preservation?”
Could you give an example of how to use “there are no moral truths” to guide your actions?
“Nothing is objectively right or wrong, therefore I will do whatever I feel like as long as I can get away with it.”
You or I might not consider this to be good moral behavior, but for a person who believed both that moral rules are only worth following if they’re objective, and that objective moral rules do not exist, it would be a reasonable conclusion to draw.
Atheism is, of course, a belief. It’s a belief that there are not any gods.
The belief that there are no objective moral rules only guides your actions to the extent that it may relieve you of constraints that you might have had if you thought there were any. But whether any other moral code is “better” than this would simply come down to a matter of value judgment.
Right. So the fact that it works as a necessary ingredient of physics is evidence of its general ability to derive true statements from other true statements.
Combining ink blobs and mating rabbits are not properly represented by the mathematical operation of addition, that’s simply an example of using the wrong math.
You asked me, several comments upthread, whether I thought the universe would be observably different if math didn’t work. The fact that not all possible mathematical statements reflect reality doesn’t get us around the answer being “yes.”
Which does not itself support your claim that it has the same epistemology as physical science.
But it’s not seen as a disproof of any mathematical claim....although it would be a disproof of a physical theory, if it mispredicted an outcome … because physics has an empirical epistemology and maths doesn’t.
Which was part of a wider point about whether maths has an empirical epistemology, which was part of a wider point about moral claims having a sui generis epistemology.
You expect reality to be different if deriving-a-true-statement-from-another-true-statement is different? But it is different. You can add or drop the Axiom of Choice. You can add or drop proof by Reductio (constructivism). Etc.
This is not a claim I ever made in the first place. The fact that we do not use empirical investigation to determine the truth of individual mathematical claims is irrelevant to my point.
There’s nothing wrong with moral data being generated by a different sort of process than we use to generate scientific data, if you can demonstrate that the process works in the first place.
Absent that kind of evidence, evidence we have in overabundance for math, you should make no such assumption.
Do we actually have any such proofs? You seem to think maths can be proven to work empoirically, but that seems to depend on cherry picking the right maths and the right physical objects.
We have an abundance of such evidence.
Can you give examples of math not working then? The fact that you can combine two blobs of ink to get one blob of ink has no bearing on the correctness of 1+1=2, because “+” is not an appropriate operator to represent mushing two objects together until they merge and then counting the merged object.
The fact that you get the wrong answers to your questions if you attempt to answer them with the wrong math is not relevant to the proposition that math works, just as the fact that using the wrong map will not help you navigate to your destination is irrelevant to the proposition that cartography works.
Do we really have an abundance? As far as I know, the non-circular vindication of popular forms of epistemollogical justification is a complex unsolved problem.
I can give examples of examples of correct maths that are also examples of incorrect maths. I can easily give examples of correct maths that are incorrect physics. “Working” has no single meaning here.
1+1=2 is a mathematical truth, period. If it doesn’t apply to inkdrops, that is a physics issue.
I amnot attempting to argue that maths does not work. I am arguing that it does not work like physics. It has a different epistemology.
It’s not that the equation doesn’t apply to inkdrops, it’s just that it doesn’t apply to doing that particular operation on inkdrops. 1+1=2 applies to inkdrops, as it does to any other physical objects, as long as you put them alongside each other and count them separately, rather than mashing them together and counting them as a unit. The operation of 1+1=2 never applies to mashing individual things together and counting how many of the mashed-together unit you have, although other qualities such as their masses will continue to be additive.
I am not arguing that mathematical truths are generated by the same epistemology as scientific data, but I am arguing that the entire edifice of math is supported by evidence, in the same manner that the edifice of the scientific method is supported by evidence, and that we do not likewise have evidence to support the edifice of an epistemology for generating objective moral truths.
What do you count as evidence in math?
You can go upwards in the tree of comments to see where I’ve discussed this already, but if you still find my meaning unclear I could answer whatever questions you might have to clarify my position.
OK. Does physical evidence of math “working” in physics count for “the entire edifice of math is supported by evidence”, or is it totally unrelated?
I would contend that it counts.
OK, thanks. I have looked through the thread again, and it looks like our views are too far apart for a forum exchange to be an adequate medium of discussing them. Or maybe I need to express them to myself clearer first.
That’s a difference that doesn’t make a difference. It remains the case that a mathematical truth is not automatically a physical truth.
It remains the case that the skill, activity, and concern of picking out the bits of maths that are relevant to a given physical situation is physics
Then what’s the difference?
And what difference is evidence making to maths? You have axioms, which are true by stipulation, and you have truth -preserving rules of inference. Logically, if you combine the two, you will generate true theorems. So where is the problem? Is the evidence supposed to remove doubt that the rules of inference are valid, or that the axioms are true?
The former. Some axioms will be applicable to reality and some will not, but the rules of inference work in either case.
Scientists tend to arrive at consensuses because they’re working with the same fundamental rules of inference and looking at the same data. Mathematicians arrive at consensuses on the consequences of given axioms, because they’re working with the same fundamental rules of inference. In both cases, we have evidence that these rules of inference actually work, which is not the case by default.
Even if we supposed that we had a system for moral reasoning which, like math, could extrapolate the consequences of axioms to their necessary conclusions (which I would argue we don’t actually have, humans are simply not good enough at natural language reasoning to pull that off with any sort of reliability,) how would we determine if any of those moral axioms apply to the real world, the way some mathematical axioms observably do?
And if some axioms are applicable to reality, then the whole edifice is supported? You mean, maths is like a rigid structure, where if one part supported, that in turn supports the other parts? That would be the case if you could logically infer axioms B and C from axiom A. But that is exactly how axioms don’t work. A set of axioms is a minimal set of assumptions: if one of your axioms is derivable from the others, it shouldn’t be in the set.
Maths doesn’t need empirical support to work as maths, since it isn’t physics, and adding empirical support doesn’t show “it works” in any exhaustive way, only that some bits of it are applicable to physical reality,
Whatever “actually works” means. If a true mathematical statement is one that is derived by standard rules of inference from standard axioms, then it could hardly fail to “work”, in that sense. OTOH, it can still fail to apply to reality, and so fail to work in that sense. (You really need to taboo “work”). But in the second case, evidence is only supporting the idea that some maths works, in the second sense. You make things rather easy by defining works to mean sometimes works...
The job of morality is to guide action, not to passively reflect facts, so the question is irrelevant.
The “standard rules of inference” could very easily simply be nonsense. Sets of rules do not help produce meaningful information by default. Out of the set of all methods of inference that could hypothetically exist, the vast majority do not produce information that is either true or consistent.
So let’s suppose that we live in a universe in which it is objectively true that there are no objective moral standards. In your view, would this be a relevant fact about morality?
In what sense? Are there alternative rules that are also truth preserving? Is truth-preservation hard to establish?
I wouldn’t say they produceinformation at all. Truth isn’t meaning, preservation isn’t production, truth preservation therefore isn’t meaning production.
We are supposed to have already chosen a set of rules that is truth-preserving.: ie, if you treat truth as a sort of numerical value, wihout any metaphysics behind it, that is assigned by fiat to your axioms, then you can show that it is preserved by using truth tables and the like (although there as some circularities there). Do you doubt that? Or is your concern showing that axioms are somehow “really” true, about something other than themselves?
It’s more metaethical. You cant use “there are not moral truths” to guide your actions at the object level..
If you count empiricism as a set of alternative rules which are truth preserving, then yes. If you’re talking about other non-empirical sets of rules, I would at least tentatively say no; formal logic is a branch of math, and natural language logic really doesn’t meet the same standards.
In that case, what if anything would you describe as the production of information?
What set of rules “we are supposed to have already chosen” are you referring to? Based on some of your statements about morality so far, I suspect I would argue that you are extrapolating from axioms that are not true in real life, much as some mathematical axioms seem to apply to the real world, while other ones do not.
Sure you can. Whether you should is an entirely separate value judgment.
If empiricism just means gathering observations, stamp collecting style, then i dont see where the truth preservation comes in. I can see where truth preservation comes into making predictions from evidence you already have, but that seems to use logical inference.
If you draw a line around an entity, then there will be information crossing that line, which is new, unpredictable, and “produced” as far as that entity is concerned.
I meant we are supposed to have already chosen a set of truth preserving mathematical and logical rules.
Could you give an example of how to use “there are no moral truths” to guide your actions?
Could you taboo “truth preservation?”
“Nothing is objectively right or wrong, therefore I will do whatever I feel like as long as I can get away with it.”
You or I might not consider this to be good moral behavior, but for a person who believed both that moral rules are only worth following if they’re objective, and that objective moral rules do not exist, it would be a reasonable conclusion to draw.
Who the problem with truthpreservation? It’s a technical term, and i gave a link
Your notion of using metaethical nihilism to guide action is analogous to treating atheism as a belief. Your actions would be unguided.
Atheism is, of course, a belief. It’s a belief that there are not any gods.
The belief that there are no objective moral rules only guides your actions to the extent that it may relieve you of constraints that you might have had if you thought there were any. But whether any other moral code is “better” than this would simply come down to a matter of value judgment.
...Gödel?
Which boils down to “math works for the situations in which it works”.
The math works regardless, it simply doesn’t apply to every situation.
A correct map of Michigan will remain correct whether you’re trying to find your way through Michigan or Taiwan.
Well, I have this thing, let’s call it htam, it says 1 + 1 = 1. Works for ink blobs perfectly well.
Actually, you know what, it works regardless, it simply doesn’t apply to every situation.
Applying maths is physics. Physical untruth can be mathematical truth and vice ver.sa.So they have different epistemologies.
I call it 2.
http://en.wikipedia.org/wiki/Two-element_Boolean_algebra
“math” is just a fancy name for “map”. Some maps do not represent any “real” places.