If I could show you an example of mathematicians running ongoing computer simulations in order to test theories (well. Test conjectures for progressively higher values), would that demonstrate otherwise to you?
And it’s not as if proofs and logic are not employed in other fields when the option is available. Isn’t the link between physics and mathematics a long-standing one, and many of the predictions of quantum theory produced on paper before they were tested?
If I could show you an example of mathematicians running ongoing computer simulations in order to test theories (well. Test conjectures for progressively higher values), would that demonstrate otherwise to you?
This happens, but the conclusion is different. No matter how many cases of an infinite-case conjecture I test, it’s not going to be accepted as proof or even particularly valid evidence that the conjecture is true. The point of doing this is more to check if there are any easy counter-examples, or to figure out what’s going on in greater detail, but then you go back and prove it.
That is evidence that a weaker conjecture (e.g. that the conjecture holds over some very huge range of numbers) is true.
And the proof verification can be seen as an empirical process. In fact it should be, given that proof verification is an experiment run on a physical machine which has limited reliability and a probability of error.
You go back and prove it if you can—and are mathematicians special in that regard, save that they deal with concepts more easily proven than most? When scientists in any field can prove something with just logic, they do. Evidence is the tiebreaker for exclusive, equivalently proven theories, and elegance the tiebreaker for exclusive, equivalently evident theories.
And that seems true for all fields labeled either a science or a form of mathematics.
When scientists in any field can prove something with just logic, they do.
I would say that when they do this, they are doing mathematics instead of science. (By the time that scientists can prove something with logic, they necessarily have a mathematical model of their field.)
I’d say it’s a little more complicated than this. In those fields where solid mathematical models have been developed, there’s usually some back-and-forth between experimentalists and theoreticians, and any particular idea is usually considered mildly suspect until both rigorous mathematical models and solid empirical backing exist. New ideas might emerge either from the math or from empirical findings.
These days in physics and chemistry the mathematical models usually seem to emerge first, but that’s not true for all fields; in astronomy, say, it’s common for observations to go unexplained or to clank along with sketchy models for quite a while before the math properly catches up.
If that’s the case, and if it is also the case that scientists prefer to use proofs and logic where available (I can admittedly only speak for myself, for whom the claim is true), then I would argue that all scientists are necessarily also mathematicians (that is to say, they practice mathematics).
And, if it is the case that mathematicians can be forced to seek inherently weaker evidence when proofs are not readily available, then I would argue that all mathematicians are necessarily also scientists (they practice science).
At that point, it seems like duplication of work to call what mathematicians and scientists do different things. Rather, they execute the same methodology on usually different subject matter (and, mind you, behave identically when given the same subject matter). You don’t have to call that methodology “the scientific method”, but what are you gonna call it otherwise?
And, if it is the case that mathematicians can be forced to seek inherently weaker evidence when proofs are not readily available
Their inherently weaker evidence still isnt empirical evidence. Computation isn;t intrinsically emprical, because a smart enough mathematician could do it in their head...they are just offloading the cognitive burden.
Fair enough, but I think that the example of mathematics was brought forth because of the thing mathematicians primarily do.
So we could rather say that “the scientific method” refers to the things scientists (and mathematicians) do when there are no proofs, which is to test ideas through experiment, and “the mathematical method” refers to proving things.
Of course, you don’t have to draw this distinction if you prefer not to; if you claim that both of these things should be called “the scientific method” then that’s also fair. But I’m pretty sure that the “Science vs. Bayes” dilemma refers only to the first thing by “Science”, since “Bayes” and “the mathematical method” don’t really compete in the same playing field.
I think that’s a workable description of the process, but using that you still have the mathematical tendency to appreciate elegance, which, on this process model, doesn’t seem like it’s in the same place as the “mathematical method” proper—since elegance becomes a concern only after things are proven.
You could argue that elegance is informal, and that this aspect of the “Science V Bayes” argument is all about trying to formalize theory elegance (in which case, it could do so across the entire process), and I think that’d be fair, but it’s not like elegance isn’t a concept already in science, though one preexisting such that it was simply not made an “official” part of “Science”.
So to try to frame this in the context of my original point, those quantum theorists who ignore an argument regarding elegance don’t strike me as being scientists limited by the bounds of their field, but scientists being human and ignoring the guidelines of their field when convenient for their biases—it’s not like a quantum physicist isn’t going to know enough math to understand how arguments regarding elegance work.
When scientists in any field can prove something with just logic, they do. Evidence is the tiebreaker
You have it backwards. Evidence is the only thing that counts. Logic is a tool to make new models, not to test them. Except in mathematics, where there is no way to test things experimentally.
If I could show you an example of mathematicians running ongoing computer simulations in order to test theories (well. Test conjectures for progressively higher values), would that demonstrate otherwise to you?
Because as Kindly notes, this happens. Mathematicians do sometimes reach for mere necessary-but-not-sufficient evidence for their claims, rather than proof. But obviously, they don’t do so when proof is more accessible—and usually, because of the subject matter mathematicians work with, it is.
There is a difference between checking the internal consistency of a simulation and gathering evidence. Scientists who use simulation calibrate the simulation with empirical measurements, and they generally are running the simulation to make predictions that have to be tested against yet more empirical measurement.
Mathematicians are just running a simulation in a vacuum. Its a very different thing.
When scientists in any field can prove something with just logic, they do. Evidence is the tiebreaker
What is an example of something a scientist can prove with ‘just logic’?
I wasn’t comparing scientists running a simulation with mathematicians running a simulation. I was comparing scientists collecting evidence that might disprove their theories with mathematicians running a simulation—because such a simulation collects data that might disprove their conjectures.
What is an example of something a scientist can prove with ‘just logic’?
We’ll need to agree on a subject who is a scientist and not a mathematician. The easiest example for me would be to use a computer scientist, but you may argue that whenever a computer scientist uses logic they’re actually functioning as a mathematician, in which case the dispute comes down to ‘what’s a mathematician’.
In the event you don’t dispute, I’d note that a lot of computer science has involved logic regarding, for instance, the nature of computation.
In the event you do dispute the status of a computer science as science, then we still have an example of scientists performing mathematics when possible, and really physicists do that too (the quantum formulas that don’t mean anything are a fine example, I think). So, to go back to my original point, it’s not like an accusation of non-elegance has to come from nowhere; those physicists are undeniably practicing math, and elegance is important there.
No matter how many cases of an infinite-case conjecture I test, it’s not going to be accepted as proof or even particularly valid evidence that the conjecture is true.
It’s incredibly common practice for mathematicians to believe that the truth of even one special case of a conjecture is evidence that the conjecture is true. For instance, the largest currently-active research program started out this way.
We must be using different meanings of the word “evidence”, because it seems under your definition consequentialist mathematicians would be completely apathetic.
I’m sorry, that made no sense to me. How about I try to restate what I was trying to say?
Mathematicians believe in mathematical results on a tiered scale: first come all the proven results, and next come all the unproven results they believe are true. Testing the first 10^6 cases of a conjecture meant to apply to all integers puts it pretty high up there in the second tier, but not as high as the first tier.
Arguably, for a result to become a theorem it has to become widely accepted by the mathematical community, at which point the proof is almost certainly correct. If I read a paper that proves a result nobody really cares about, I’m more careful about believing it.
Of course, you can assume a conjecture and use it to prove a theorem; then the theorem you proved is a valid result of the form “if this conjecture is true, then”. However, a stronger result that doesn’t assume the conjecture is better.
So when you said what I quoted in the great-grandparent, that is, “[10^6 cases aren’t] going to be accepted as … particularly valid evidence”, you meant that it would be accepted as evidence, that is, it belongs to your second tier of belief? That’s what’s bothering me.
Sorry, I guess at that point I was just thinking of the first tier of belief when making that comment.
But I think it’s also true that most notable conjectures have, in addition to verification of some cases, some sort of handwavy reason why we would believe them, such as a sketch of a proof but with many holes in it, and this is at least important as the other kind of evidence.
Here is the difference: the superstring theory is a reasonably good mathematical model which predicts a spacetime with 10 or 11-dimensions on purely mathematical grounds. It also predicts that particles should come in pairs (quarks+squarks). Despite its internal self-consistency, it’s not a good model of the world we live in. Whether mathematicians use the scientific method depends on your definition of the scientific method (a highly contested issue on the relevant wikipedia page). Feel free to give your definition and we can go from there.
I feel this reply I made captures the link between proof, evidence, and elegance, in both scientific and mathematical fields.
That is to say, where proof is equivalent for two mutually exclusive theories (because sometimes things are proven logically outside mathematics, and not everything in mathematics are proven), evidence is used as a tiebreaker.
And where evidence is equivalent for two mutually exclusive theories (requiring of course that proof also be equivalent), elegance is used as a tiebreaker.
Here is the difference: the superstring theory is a reasonably good mathematical model which predicts a spacetime with 10 or 11-dimensions on purely mathematical grounds.
Not quite. More like abstractly physical gorunds...combining various symmetry principles from preceding theories.
Despite its internal self-consistency, it’s not a good model of the world we live in.
Not quite. it doesn’t predict a single world that is different. It predicts a landscape in which our world
may be located with difficulty.
If I could show you an example of mathematicians running ongoing computer simulations in order to test theories (well. Test conjectures for progressively higher values), would that demonstrate otherwise to you?
And it’s not as if proofs and logic are not employed in other fields when the option is available. Isn’t the link between physics and mathematics a long-standing one, and many of the predictions of quantum theory produced on paper before they were tested?
This happens, but the conclusion is different. No matter how many cases of an infinite-case conjecture I test, it’s not going to be accepted as proof or even particularly valid evidence that the conjecture is true. The point of doing this is more to check if there are any easy counter-examples, or to figure out what’s going on in greater detail, but then you go back and prove it.
That is evidence that a weaker conjecture (e.g. that the conjecture holds over some very huge range of numbers) is true.
And the proof verification can be seen as an empirical process. In fact it should be, given that proof verification is an experiment run on a physical machine which has limited reliability and a probability of error.
You go back and prove it if you can—and are mathematicians special in that regard, save that they deal with concepts more easily proven than most? When scientists in any field can prove something with just logic, they do. Evidence is the tiebreaker for exclusive, equivalently proven theories, and elegance the tiebreaker for exclusive, equivalently evident theories.
And that seems true for all fields labeled either a science or a form of mathematics.
Hmmmm}...
I would say that when they do this, they are doing mathematics instead of science. (By the time that scientists can prove something with logic, they necessarily have a mathematical model of their field.)
I’d say it’s a little more complicated than this. In those fields where solid mathematical models have been developed, there’s usually some back-and-forth between experimentalists and theoreticians, and any particular idea is usually considered mildly suspect until both rigorous mathematical models and solid empirical backing exist. New ideas might emerge either from the math or from empirical findings.
These days in physics and chemistry the mathematical models usually seem to emerge first, but that’s not true for all fields; in astronomy, say, it’s common for observations to go unexplained or to clank along with sketchy models for quite a while before the math properly catches up.
If that’s the case, and if it is also the case that scientists prefer to use proofs and logic where available (I can admittedly only speak for myself, for whom the claim is true), then I would argue that all scientists are necessarily also mathematicians (that is to say, they practice mathematics).
And, if it is the case that mathematicians can be forced to seek inherently weaker evidence when proofs are not readily available, then I would argue that all mathematicians are necessarily also scientists (they practice science).
At that point, it seems like duplication of work to call what mathematicians and scientists do different things. Rather, they execute the same methodology on usually different subject matter (and, mind you, behave identically when given the same subject matter). You don’t have to call that methodology “the scientific method”, but what are you gonna call it otherwise?
Their inherently weaker evidence still isnt empirical evidence. Computation isn;t intrinsically emprical, because a smart enough mathematician could do it in their head...they are just offloading the cognitive burden.
Fair enough, but I think that the example of mathematics was brought forth because of the thing mathematicians primarily do.
So we could rather say that “the scientific method” refers to the things scientists (and mathematicians) do when there are no proofs, which is to test ideas through experiment, and “the mathematical method” refers to proving things.
Of course, you don’t have to draw this distinction if you prefer not to; if you claim that both of these things should be called “the scientific method” then that’s also fair. But I’m pretty sure that the “Science vs. Bayes” dilemma refers only to the first thing by “Science”, since “Bayes” and “the mathematical method” don’t really compete in the same playing field.
I think that’s a workable description of the process, but using that you still have the mathematical tendency to appreciate elegance, which, on this process model, doesn’t seem like it’s in the same place as the “mathematical method” proper—since elegance becomes a concern only after things are proven.
You could argue that elegance is informal, and that this aspect of the “Science V Bayes” argument is all about trying to formalize theory elegance (in which case, it could do so across the entire process), and I think that’d be fair, but it’s not like elegance isn’t a concept already in science, though one preexisting such that it was simply not made an “official” part of “Science”.
So to try to frame this in the context of my original point, those quantum theorists who ignore an argument regarding elegance don’t strike me as being scientists limited by the bounds of their field, but scientists being human and ignoring the guidelines of their field when convenient for their biases—it’s not like a quantum physicist isn’t going to know enough math to understand how arguments regarding elegance work.
You have it backwards. Evidence is the only thing that counts. Logic is a tool to make new models, not to test them. Except in mathematics, where there is no way to test things experimentally.
Your claim leads me back to my earlier statement.
Because as Kindly notes, this happens. Mathematicians do sometimes reach for mere necessary-but-not-sufficient evidence for their claims, rather than proof. But obviously, they don’t do so when proof is more accessible—and usually, because of the subject matter mathematicians work with, it is.
There is a difference between checking the internal consistency of a simulation and gathering evidence. Scientists who use simulation calibrate the simulation with empirical measurements, and they generally are running the simulation to make predictions that have to be tested against yet more empirical measurement.
Mathematicians are just running a simulation in a vacuum. Its a very different thing.
What is an example of something a scientist can prove with ‘just logic’?
I wasn’t comparing scientists running a simulation with mathematicians running a simulation. I was comparing scientists collecting evidence that might disprove their theories with mathematicians running a simulation—because such a simulation collects data that might disprove their conjectures.
We’ll need to agree on a subject who is a scientist and not a mathematician. The easiest example for me would be to use a computer scientist, but you may argue that whenever a computer scientist uses logic they’re actually functioning as a mathematician, in which case the dispute comes down to ‘what’s a mathematician’.
In the event you don’t dispute, I’d note that a lot of computer science has involved logic regarding, for instance, the nature of computation.
In the event you do dispute the status of a computer science as science, then we still have an example of scientists performing mathematics when possible, and really physicists do that too (the quantum formulas that don’t mean anything are a fine example, I think). So, to go back to my original point, it’s not like an accusation of non-elegance has to come from nowhere; those physicists are undeniably practicing math, and elegance is important there.
Has anyone come up with a decent model by mechanically applying a logical procedure?
It’s incredibly common practice for mathematicians to believe that the truth of even one special case of a conjecture is evidence that the conjecture is true. For instance, the largest currently-active research program started out this way.
To the extent that the research is worth pursuing, yes; but no mathematician treats such a conjecture in the same way as a proven result.
We must be using different meanings of the word “evidence”, because it seems under your definition consequentialist mathematicians would be completely apathetic.
I’m sorry, that made no sense to me. How about I try to restate what I was trying to say?
Mathematicians believe in mathematical results on a tiered scale: first come all the proven results, and next come all the unproven results they believe are true. Testing the first 10^6 cases of a conjecture meant to apply to all integers puts it pretty high up there in the second tier, but not as high as the first tier.
Arguably, for a result to become a theorem it has to become widely accepted by the mathematical community, at which point the proof is almost certainly correct. If I read a paper that proves a result nobody really cares about, I’m more careful about believing it.
Of course, you can assume a conjecture and use it to prove a theorem; then the theorem you proved is a valid result of the form “if this conjecture is true, then”. However, a stronger result that doesn’t assume the conjecture is better.
So when you said what I quoted in the great-grandparent, that is, “[10^6 cases aren’t] going to be accepted as … particularly valid evidence”, you meant that it would be accepted as evidence, that is, it belongs to your second tier of belief? That’s what’s bothering me.
Sorry, I guess at that point I was just thinking of the first tier of belief when making that comment.
But I think it’s also true that most notable conjectures have, in addition to verification of some cases, some sort of handwavy reason why we would believe them, such as a sketch of a proof but with many holes in it, and this is at least important as the other kind of evidence.
Here is the difference: the superstring theory is a reasonably good mathematical model which predicts a spacetime with 10 or 11-dimensions on purely mathematical grounds. It also predicts that particles should come in pairs (quarks+squarks). Despite its internal self-consistency, it’s not a good model of the world we live in. Whether mathematicians use the scientific method depends on your definition of the scientific method (a highly contested issue on the relevant wikipedia page). Feel free to give your definition and we can go from there.
I feel this reply I made captures the link between proof, evidence, and elegance, in both scientific and mathematical fields.
That is to say, where proof is equivalent for two mutually exclusive theories (because sometimes things are proven logically outside mathematics, and not everything in mathematics are proven), evidence is used as a tiebreaker.
And where evidence is equivalent for two mutually exclusive theories (requiring of course that proof also be equivalent), elegance is used as a tiebreaker.
Not quite. More like abstractly physical gorunds...combining various symmetry principles from preceding theories.
Not quite. it doesn’t predict a single world that is different. It predicts a landscape in which our world may be located with difficulty.