Surely, there must be some fact of the matter about what the ratio of a circle’s circumference to its diameter is?
This is exactly the issue at hand. You believe in external mathematical “facts”, ideal platonic objects. The mathematical territory. This is a useful belief at times, but not in this case, as it gets in the way of making otherwise obvious predictions about observations, such as “how likely that a randomly picked digit of π is zero, once it is picked, but not yet calculated?”
Well, let me put it another way. Suppose that I calculate the 98,765th digit of π. And my friend Hasan, who lives on the other side of the world, also, separately, calculates the 98,765th digit of π. Can we get different results? (Other than by making some mistake in writing the code that does the calculation, or some such.) Is that a thing that can happen? What is the probability of the 98,765th digit of π being one thing when calculated by one person, but something else when calculated by someone else, elsewhere? (And if nonzero, how far does this go—could the 1,500th digit of π vary from person to person? The 220th? The 30th? The 3rd?!)
If you say that this sort of thing can happen, well, then you’re certainly saying something novel and strange. I guess all I have to say to that is “[citation needed]”. But, if (as seems more likely) you agree that such a thing cannot happen, then my question is: just what exactly is it that makes the 98,765th of π be the same thing when calculated by me, or by Hasan, or by anyone else? Whatever that thing is, what is wrong with calling it “a fact of the matter about what the 98,765th digit of π is”?
You seem to be conflating two different questions:
What is your best estimate of probability of the currently unknown to you 98,765th digit of π coming out zero, once someone calculates it?
and
What is your best estimate of probability of the 98,765th digit of π calculated by two different people being different?
Once enough people reliably do the same calculation (or if there is another reliable way to perform the observation of the 98,765th digit of π), then it can be added to the list of performed observations and, if needed used to predict future observations.
just what exactly is it that makes the 98,765th of π be the same thing when calculated by me, or by Hasan, or by anyone else? Whatever that thing is, what is wrong with calling it “a fact of the matter about what the 98,765th digit of π is”
This goes back to realism vs anti-realism, not anything I had invented. Anti-realism is a self-consistent epistemology, it pops up in many areas independently. According to Wikipedia, in science an example of it in science is instrumentalism, and in math it is intuitionism: “there are no non-experienced mathematical truths”.
There is no difference between logical uncertainty and environmental uncertainty in anti-realism. OP seems to have reinvented the juxtaposition of realism and anti-realism in the setting of the probability theory, calling it “perfect Bayesianism” and “subjective Bayesianism” respectively. And “perfect Bayesianism” runs into trouble with logical vs environmental uncertainties, because of the extra (and unnecessary, in the anti-realist view) postulate of objective reality.
I still don’t think you’ve answered Said’s question. The question is whether two people can observe different values of pi. Or, to put it differently, why is it that, whenever anyone computes a value of pi, it seems to come out to the same value (3.14159...). Doesn’t that indicate that there is some kind of objective reality, to which our mathematics corresponds?
One of the questions that Wigner brings up in The Unreasonable Effectiveness of Mathematics in the Natural Sciences is why does our math work so well at predicting the future? I would put the same question to you, but in a more general form. If there is no such thing as non-experienced mathematical truths, then why does everyone’s experience of mathematical truths seem to be the same?
Doesn’t that indicate that there is some kind of objective reality, to which our mathematics corresponds?
A reality behind repeatable observations is a good model, as long as it works. My point is that it doesn’t always work, like in the confusion about logical uncertainty.
And I disagree with the assumptions behind the Wigner’s question, “why does our math work so well at predicting the future?”, specifically that math’s effectiveness is “unreasonable”. Human and animal brains do complicated calculations all the time in real time to get through life, like solving what amounts to non-linear partial differential equations to even get a bite of food into your mouth. Just because it is subconscious, it is no less of a math than proving theorems. What most humans mean by math is constructing conscious, not subconscious meta-models and using them in multiple contexts. But we subconscious meta-modeling like this all the time in other areas of human experience, so my answer to Wigner’s question is “you are committing a mind projection fallacy, the apparently unreasonable effectiveness of mathematics is a statement about human mind, not about the world”.
If there is no such thing as non-experienced mathematical truths, then why does everyone’s experience of mathematical truths seem to be the same?
In general, however, your questions about the intuitionist approach to math is best directed to professional mathematicians who are actually intuitionists, though.
Human and animal brains do complicated calculations all the time in real time to get through life, like solving what amounts to non-linear partial differential equations to even get a bite of food into your mouth. Just because it is subconscious, it is no less of a math than proving theorems.
I agree. So if there is no “objective” reality, apart from that which we experience, then why is it that we all seem to experience the same reality? When I shoot a basketball, or hit a tennis ball, both I and the referee see the same trajectory and are in approximate agreement about where the ball lands. When I lift a piece of food to my mouth and eat it, it would surprise me if someone across the table said that they saw it spill from my fork and stain my shirt.
In the absence of an external reality, why is it that everyone’s model of the world appears to be in such concordance with everyone else’s?
So if there is no “objective” reality, apart from that which we experience, then why is it that we all seem to experience the same reality?
I am not saying that there is no objective reality, just that I am agnostic about it. In the example you describe, it is a useful meta-model, though not all the time. You may notice that, despite a video review and slow motion hi-res cameras, fans of different teams still argue about what happened, and the final decision is in the hands of a referee. You and your partner (especially ex partner) may disagree about “what really happened” and there is often no way to tell “who is right”. One instead has to accept that what one person experienced is not necessarily what another did, and, at least instrumentally, arguing about whose reality is the “true” is likely to be not useful at all. One may as well accept the model where somewhat different things happened to different actors.
In the absence of an external reality, why is it that everyone’s model of the world appears to be in such concordance with everyone else’s?
Does it? Who won the World War II, Americans, British or Russians? Is Trump a hero or a villain? Did Elon Musk disclose material information or not in his tweets? Do mathematical infinities exist? Are the laws of physics invented or discovered? Was Jesus a son of God? The list of disagreements about “objective reality” is endless. Sure, there is some “concordance” between different people’s views of the world, but it is much less strong than one naively assumes.
The examples you use reinforce my point. We argue about extremely fine details. When supporters of opposing teams argue over whether a point was or was not scored, they’re disputing whether the ball was here or there by a few millimeters. You won’t find very many people arguing that actually, the ball was clear on the other side of the field and in reality, the disputed point is one that would have been scored by the other team.
Similarly, we might argue about whether the British, Americans or Russians were primarily responsible for the United Nations’ victory in World War 2, but I don’t think you’ll find very many people arguing that actually it was the Italians who won World War 2.
The fact that our perceptions of reality match each other 99.999% of the time, to me, indicates that there’s something out there that exists regardless of whether I perceive it or not. I call that “reality”.
I can see your point, and it’s the one most people implicitly accept. Observations are predictable, therefore there is a shared reality out there generating those observations. It works most of the time. But in the edge cases (or “extremely fine details”) this implicit assumption breaks down. Like in the case of “objective mathematical facts waiting to be discovered”, such as the 98,765th of π before you measure it. So why insist on applying this assumption outside of its realm of applicability? Isn’t it sort of like insisting that if you shoot a bullet from a ship moving with nearly the speed of light, it will travel faster than light?
You seem to be saying that “external shared reality” is an approximation in the same way that Newtonian mechanics is an approximation for Einsteinian relativity. That’s fine. So what is “external shared reality” an approximation of? Just what exactly is out there generating inputs to my senses, and by what mechanism does it remain in sync with everyone else (approximately)?
Just what exactly is out there generating inputs to my senses, and by what mechanism does it remain in sync with everyone else (approximately)?
Sometimes the “out there” can be modeled as a shared reality, sure. The key word is “modeled”. Sometimes this model is not a good one. If you insist on privileging one model over all others to be the true objective external reality valid everywhere, you pay the price where it fails. Like in the OP’s case.
Having read through the above discussion, I don’t think you have distinguished between the claim that there are mathematical entities, and the claim that there are mathematical facts. The latter can mean nothing more than different mathematicians will find the same solutions to a given problem, which you accept. Call the second claim epistemological realism, and the first metaphysical realism. To argue that convergence on a set of facts can only be, or be explained by, form of metaphysical realism is to give to much credence to realism. Metaphysical realism about mathematical entities , Platonism, is much more controversial than realism about physical bodies.
By “good” I mean (as always) “fitting the available observations and producing accurate predictions”. In the OP’s case of the 98,765th digit of π, the model is that “A randomly picked digit is uniformly distributed” and it is a “good” (i.e. accurate) one.
There’s a puzzle about how probability theory would apply would apply to something that’s basically determinate, but the question of how randomly selected digits of pi are distributed isn’t it, because the process of picking a digit randomly bring indeterminacy in.
People pose the problem with a specific digit to make the problem determinate, and focus on the paradoxical aspect.
The paradox only arises if you ignore the view I’ve been presenting. The 98,765th digit of π is a random digit in the same way that a 98,765th reading of rand() is. Until you do some work to measure it, it’s not determined.
It is determined in the sense of having only one possible value. The same applies to a call to rand() ,so long as it is a deterministic PRNG. We don’t know what the answer is , until we have done some work, in either case, but that doesn’t mean anything indeterministic is going on. Determinism is defined in terms of inevitability, ie. lack of possible alternatives. We do not regard the future as undeterminedjust because it has not happened yet.
Determinism is defined in terms of inevitability, ie. lack of possible alternatives. We do not regard the future as undetermined just because it has not happened yet.
I don’t argue with that, in fact, the statement above makes my point: there is no difference between an as-yet-unknown to you (but predetermined) digit of pi and anything else that is not yet known to you, like the way a coin lands when you flip it.
It doens’t make your point, since I don’t agree with it.
Given any degree of realism, you can differentiate between determined but unknown things and undetermined things.
Well, you’re an anti realist. But that doesn’t give you the right to interpret what other people, if there are any other people, are saying in anti-realist terms.
This is exactly the issue at hand. You believe in external mathematical “facts”, ideal platonic objects. The mathematical territory. This is a useful belief at times, but not in this case, as it gets in the way of making otherwise obvious predictions about observations, such as “how likely that a randomly picked digit of π is zero, once it is picked, but not yet calculated?”
Well, let me put it another way. Suppose that I calculate the 98,765th digit of π. And my friend Hasan, who lives on the other side of the world, also, separately, calculates the 98,765th digit of π. Can we get different results? (Other than by making some mistake in writing the code that does the calculation, or some such.) Is that a thing that can happen? What is the probability of the 98,765th digit of π being one thing when calculated by one person, but something else when calculated by someone else, elsewhere? (And if nonzero, how far does this go—could the 1,500th digit of π vary from person to person? The 220th? The 30th? The 3rd?!)
If you say that this sort of thing can happen, well, then you’re certainly saying something novel and strange. I guess all I have to say to that is “[citation needed]”. But, if (as seems more likely) you agree that such a thing cannot happen, then my question is: just what exactly is it that makes the 98,765th of π be the same thing when calculated by me, or by Hasan, or by anyone else? Whatever that thing is, what is wrong with calling it “a fact of the matter about what the 98,765th digit of π is”?
You seem to be conflating two different questions:
What is your best estimate of probability of the currently unknown to you 98,765th digit of π coming out zero, once someone calculates it?
and
What is your best estimate of probability of the 98,765th digit of π calculated by two different people being different?
Once enough people reliably do the same calculation (or if there is another reliable way to perform the observation of the 98,765th digit of π), then it can be added to the list of performed observations and, if needed used to predict future observations.
This goes back to realism vs anti-realism, not anything I had invented. Anti-realism is a self-consistent epistemology, it pops up in many areas independently. According to Wikipedia, in science an example of it in science is instrumentalism, and in math it is intuitionism: “there are no non-experienced mathematical truths”.
There is no difference between logical uncertainty and environmental uncertainty in anti-realism. OP seems to have reinvented the juxtaposition of realism and anti-realism in the setting of the probability theory, calling it “perfect Bayesianism” and “subjective Bayesianism” respectively. And “perfect Bayesianism” runs into trouble with logical vs environmental uncertainties, because of the extra (and unnecessary, in the anti-realist view) postulate of objective reality.
I still don’t think you’ve answered Said’s question. The question is whether two people can observe different values of pi. Or, to put it differently, why is it that, whenever anyone computes a value of pi, it seems to come out to the same value (3.14159...). Doesn’t that indicate that there is some kind of objective reality, to which our mathematics corresponds?
One of the questions that Wigner brings up in The Unreasonable Effectiveness of Mathematics in the Natural Sciences is why does our math work so well at predicting the future? I would put the same question to you, but in a more general form. If there is no such thing as non-experienced mathematical truths, then why does everyone’s experience of mathematical truths seem to be the same?
A reality behind repeatable observations is a good model, as long as it works. My point is that it doesn’t always work, like in the confusion about logical uncertainty.
And I disagree with the assumptions behind the Wigner’s question, “why does our math work so well at predicting the future?”, specifically that math’s effectiveness is “unreasonable”. Human and animal brains do complicated calculations all the time in real time to get through life, like solving what amounts to non-linear partial differential equations to even get a bite of food into your mouth. Just because it is subconscious, it is no less of a math than proving theorems. What most humans mean by math is constructing conscious, not subconscious meta-models and using them in multiple contexts. But we subconscious meta-modeling like this all the time in other areas of human experience, so my answer to Wigner’s question is “you are committing a mind projection fallacy, the apparently unreasonable effectiveness of mathematics is a statement about human mind, not about the world”.
In general, however, your questions about the intuitionist approach to math is best directed to professional mathematicians who are actually intuitionists, though.
I agree. So if there is no “objective” reality, apart from that which we experience, then why is it that we all seem to experience the same reality? When I shoot a basketball, or hit a tennis ball, both I and the referee see the same trajectory and are in approximate agreement about where the ball lands. When I lift a piece of food to my mouth and eat it, it would surprise me if someone across the table said that they saw it spill from my fork and stain my shirt.
In the absence of an external reality, why is it that everyone’s model of the world appears to be in such concordance with everyone else’s?
I am not saying that there is no objective reality, just that I am agnostic about it. In the example you describe, it is a useful meta-model, though not all the time. You may notice that, despite a video review and slow motion hi-res cameras, fans of different teams still argue about what happened, and the final decision is in the hands of a referee. You and your partner (especially ex partner) may disagree about “what really happened” and there is often no way to tell “who is right”. One instead has to accept that what one person experienced is not necessarily what another did, and, at least instrumentally, arguing about whose reality is the “true” is likely to be not useful at all. One may as well accept the model where somewhat different things happened to different actors.
Does it? Who won the World War II, Americans, British or Russians? Is Trump a hero or a villain? Did Elon Musk disclose material information or not in his tweets? Do mathematical infinities exist? Are the laws of physics invented or discovered? Was Jesus a son of God? The list of disagreements about “objective reality” is endless. Sure, there is some “concordance” between different people’s views of the world, but it is much less strong than one naively assumes.
The examples you use reinforce my point. We argue about extremely fine details. When supporters of opposing teams argue over whether a point was or was not scored, they’re disputing whether the ball was here or there by a few millimeters. You won’t find very many people arguing that actually, the ball was clear on the other side of the field and in reality, the disputed point is one that would have been scored by the other team.
Similarly, we might argue about whether the British, Americans or Russians were primarily responsible for the United Nations’ victory in World War 2, but I don’t think you’ll find very many people arguing that actually it was the Italians who won World War 2.
The fact that our perceptions of reality match each other 99.999% of the time, to me, indicates that there’s something out there that exists regardless of whether I perceive it or not. I call that “reality”.
I can see your point, and it’s the one most people implicitly accept. Observations are predictable, therefore there is a shared reality out there generating those observations. It works most of the time. But in the edge cases (or “extremely fine details”) this implicit assumption breaks down. Like in the case of “objective mathematical facts waiting to be discovered”, such as the 98,765th of π before you measure it. So why insist on applying this assumption outside of its realm of applicability? Isn’t it sort of like insisting that if you shoot a bullet from a ship moving with nearly the speed of light, it will travel faster than light?
You seem to be saying that “external shared reality” is an approximation in the same way that Newtonian mechanics is an approximation for Einsteinian relativity. That’s fine. So what is “external shared reality” an approximation of? Just what exactly is out there generating inputs to my senses, and by what mechanism does it remain in sync with everyone else (approximately)?
Sometimes the “out there” can be modeled as a shared reality, sure. The key word is “modeled”. Sometimes this model is not a good one. If you insist on privileging one model over all others to be the true objective external reality valid everywhere, you pay the price where it fails. Like in the OP’s case.
Having read through the above discussion, I don’t think you have distinguished between the claim that there are mathematical entities, and the claim that there are mathematical facts. The latter can mean nothing more than different mathematicians will find the same solutions to a given problem, which you accept. Call the second claim epistemological realism, and the first metaphysical realism. To argue that convergence on a set of facts can only be, or be explained by, form of metaphysical realism is to give to much credence to realism. Metaphysical realism about mathematical entities , Platonism, is much more controversial than realism about physical bodies.
“Sometimes this model is not a good one.”
What do you mean by “good” here? And, given some definitiin of good, what alternative model is better in that sort of situation?
By “good” I mean (as always) “fitting the available observations and producing accurate predictions”. In the OP’s case of the 98,765th digit of π, the model is that “A randomly picked digit is uniformly distributed” and it is a “good” (i.e. accurate) one.
..isn’t a random digit, it’s the 98,765th digit.
There’s a puzzle about how probability theory would apply would apply to something that’s basically determinate, but the question of how randomly selected digits of pi are distributed isn’t it, because the process of picking a digit randomly bring indeterminacy in.
People pose the problem with a specific digit to make the problem determinate, and focus on the paradoxical aspect.
The paradox only arises if you ignore the view I’ve been presenting. The 98,765th digit of π is a random digit in the same way that a 98,765th reading of rand() is. Until you do some work to measure it, it’s not determined.
It is determined in the sense of having only one possible value. The same applies to a call to rand() ,so long as it is a deterministic PRNG. We don’t know what the answer is , until we have done some work, in either case, but that doesn’t mean anything indeterministic is going on. Determinism is defined in terms of inevitability, ie. lack of possible alternatives. We do not regard the future as undeterminedjust because it has not happened yet.
I don’t argue with that, in fact, the statement above makes my point: there is no difference between an as-yet-unknown to you (but predetermined) digit of pi and anything else that is not yet known to you, like the way a coin lands when you flip it.
It doens’t make your point, since I don’t agree with it.
Given any degree of realism, you can differentiate between determined but unknown things and undetermined things.
Well, you’re an anti realist. But that doesn’t give you the right to interpret what other people, if there are any other people, are saying in anti-realist terms.
Right, never mind, for a moment what your discourse style is. Disengaging.