= 27d567bc2d48d9f40e9ccc20ea370b15)
Canberra, Australia.
In response to
Your intuitions are not magic
Yet our brains assume that we hear about all those disasters [we read about in the newspaper] because we've personally witnessed them, and that the distribution of disasters in the newspapers therefore reflects the distribution of disasters in the real world.
Even if we had personally witnessed them, that wouldn't, in itself, be any reason to assume that they are representative of things in general. The representativeness of any data is always something that can be critically assessed.
In response to
The two meanings of mathematical terms
Or: "Physics is not Math"
This seems to be a common response - Tyrrell_McAllister said something similar:
I think that your distinction is really just the distinction between physics and mathematics.
I take that distinction as meaning that a precise maths statement isn't necessarily reflecting reality like physics does. That is not really my point.
For one thing, my point is about any applied maths, regardless of domain. That maths could be used in physics, biology, economics, engineering, computer science, or even the humanities.
But more importantly, my point concerns what you think the equations are about, and how you can be mistaken about that, even in physics.
The following might help clarify.
A successful test of a mathematical theory against reality means that it accurately describes some aspect of reality. But a successful test doesn't necessarily mean it accurately describes what you think it does.
People successfully tested the epicycles theory's predictions about the movement of the planets and the stars. They tended to think that this showed that the planets and stars were carried around on the specified configuration of rotating circles, but all it actually showed was that the points of light in the sky followed the paths the theory predicted.
They were committing a mind projection 'fallacy' - their eyes were looking at points of light but they were 'seeing' planets and stars embedded in spheres.
The way people interpreted those successful predictions made it very hard to criticise the epicycles theory.
In response to
The two meanings of mathematical terms
Please see this previous comment of mine.
The point here is that it "1+1=2" should not be taken as a statement about physical reality, unless and until we have agreed (explicitly!) on a specific model of the world -- that is, a specific physical interpretation of those mathematical terms. If that model later turns out not to correspond to reality, that's what we say; we don't say that the mathematics was incorrect.
Thus, examples of things not to say:
"Relativity disproves Euclidean geometry."
"Quantum mechanics disproves classical logic"
"I am an infinite set atheist - have you ever actually seen an infinite set?"
I fully agree, and this is completely in line with the points I was trying to make.
In response to
The two meanings of mathematical terms
To add to what others have already commented...
It is theoretically possible to accurately describe the motions of celestial bodies using epicycles, though one might need infinite epicycles, and epicycles would themselves need to be on epicycles. If you think there's something wrong with the math, it won't be in its inability to describe the motion of celestial bodies. Rather, feasibility, simplicity, usefulness, and other such concerns will likely be factors in it.
While 'accurate' and 'precise' are used as synonyms in ordinary language, please never use them that way when talking technically about the meanings of words. They are very useful jargon.
Similarly, please never use 'begs the question' or any form of it when not referring to the logical fallacy.
It is theoretically possible to accurately describe the motions of celestial bodies using epicycles, though one might need infinite epicycles, and epicycles would themselves need to be on epicycles. If you think there's something wrong with the math, it won't be in its inability to describe the motion of celestial bodies.
But I don't think there's anything "wrong with the math" - I even said precisely that:
A believer in epicycles would likely have thought that it must have been correct because it gave mathematically correct answers. And it actually did . Epicycles actually did precisely calculate the positions of the stars and planets (not absolutely perfectly, but in principle the theory could have been adjusted to give perfectly precise results). If the mathematics was right, how could it be wrong?
.
While 'accurate' and 'precise' are used as synonyms in ordinary language, please never use them that way when talking technically about the meanings of words.
I was trying to talk about how people actually use them, and one of the things I was suggesting is that people do actually tend to treat them as synonymous.
Similarly, please never use 'begs the question' or any form of it when not referring to the logical fallacy.
Isn't this a little picky? The way I used 'begs the question', in the sense of 'raises the question', is fairly common usage. Language is constantly evolving and if you wanted to claim that people only should use terms and phrases in line with their original meanings you'd have throw away most language.
In response to
The two meanings of mathematical terms
"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."
-- Albert Einstein
As far as I can see, that's just an acknowledgement that we can't know anything for certain -- so we can't be certain of any 'laws', and any claim of certainty is invalid.
I was arguing that any applied maths term has two types of meanings -- one 'internal to' the equations and an 'external' ontological one, concerning what it represents -- and that a precise 'internal' meaning does not imply a precise 'external' meaning, even though 'precision' is often only thought of in terms of the first type of meaning.
I don't see how that relates in any way to the question of absolute certainty. Is there some relationship I'm missing here?
In response to
comment
by
on
The Terrible, Horrible, No Good, Very Bad Truth About Morality and What To Do About It
Indeed, and that is why it's wrong to say that attempts to rationally justify statements about reality are "almost certainly going to produce an ad hoc Just-So Story".
That's not what I said.
I have no interest in helping to generate a Gish Explosion. Please confine yourself to addressing arguments I actually make, rather than straw men.
In response to
comment
by
on
The Terrible, Horrible, No Good, Very Bad Truth About Morality and What To Do About It
I'm not trying to be a jerk. Let me try to explain things, as I don't think I communicated my point very clearly.
Just to start off, the quoted text is something you said.
But perhaps you are saying that the sentence I've embedded it in does not reflect what any thing you said? If so, it's not mean to - it's describing the point I was making, and to which your response included that quoted text.
Essentially, my last comment was trying to point out what I'd originally said had been misinterpreted in the Just-So Story bit, even though I didn't do a great job of making this clear. Of course you may argue that you didn't misinterpret me, but I certainly wasn't trying to put words into anyones mouth.
In response to
comment
by
on
The Terrible, Horrible, No Good, Very Bad Truth About Morality and What To Do About It
that implies that the only correct intuition is one you can immediately rationally justify.
Wrong. An intuition is correct if it matches reality.
Accepting an intuition is only rational if it can be rationally justified, in which case the intuition isn't needed, is it?
science is basically a means to determine whether initial intuitions are true.
No, science is a methodology to determine whether an assertion about reality should be discarded. If it merely dealt with initial intuitions, it's usefulness would be exhausted once the supply of initial intuitions had been run through.
In response to
comment
by
on
The Terrible, Horrible, No Good, Very Bad Truth About Morality and What To Do About It
An intuition is correct if it matches reality.
Indeed, and that is why it's wrong to say that attempts to rationally justify statements about reality are "almost certainly going to produce an ad hoc Just-So Story".
science is basically a means to determine whether initial intuitions are true.
No, science is a methodology to determine whether an assertion about reality should be discarded. If it merely dealt with initial intuitions, it's usefulness would be exhausted once the supply of initial intuitions had been run through.
I'm not sure what the second sentence there is taking "initial intuitions" to mean, but I don't think there's any substantial disagreement between our statements.
The two meanings of mathematical terms
[edit: sorry, the formatting of links and italics in this is all screwy. I've tried editing both the rich-text and the HTML and either way it looks ok while i'm editing it but the formatted terms either come out with no surrounding spaces or two surrounding spaces]
In the latest Rationality Quotes thread, CronoDAS quoted Paul Graham:
It would not be a bad definition of math to call it the study of terms that have precise meanings.
Sort of. I started writing a this as a reply to that comment, but it grew into a post.
We've all heard of the story of epicycles and how before Copernicus came along the movement of the stars and planets were explained by the idea of them being attached to rotating epicycles, some of which were embedded within other larger, rotating epicycles (I'm simplifying the terminology a little here).
As we now know, the Epicycles theory was completely wrong. The stars and planets were not at the distances from earth posited by the theory, or of the size presumed by it, nor were they moving about on some giant clockwork structure of rings.
In the theory of Epicycles the terms had precise mathematical meanings. The problem was that what the terms were meant to represent in reality were wrong. The theory involved applied mathematical statements, and in any such statements the terms don’t just have their mathematical meaning -- what the equations say about them -- they also have an ‘external’ meaning concerning what they’re supposed to represent in or about reality.
Lets consider these two types of meanings. The mathematical, or ‘internal’, meaning of a statement like ‘1 + 1 = 2’ is very precise. ‘1 + 1’ is defined as ‘2’, so ‘1 + 1 = 2’ is pretty much the pre-eminent fact or truth. This is why mathematical truth is usually given such an exhaulted place. So far so good with saying that mathematics is the study of terms with precise meanings.
But what if ‘1 + 1 = 2’ happens to be used to describe something in reality? Each of the terms will then take on a second meaning -- concerning what they are meant to be representing in reality. This meaning lies outside the mathematical theory, and there is no guarantee that it is accurate.
The problem with saying that mathematics is the study of terms with precise meanings is that it’s all to easy to take this as trivially true, because the terms obviously have a precise mathematical sense. It’s easy to overlook the other type of meaning, to think there is just the meaning of the term, and that there is just the question of the precision of their meanings. This is why we get people saying "numbers don’t lie".
‘Precise’ is a synonym for "accurate" and "exact" and it is characterized by "perfect conformity to fact or truth" (according to WordNet). So when someone says that mathematics is the study of terms with precise meanings, we have a tendancy to take it as meaning it’s the study of things that are accurate and true. The problem with that is, mathematical precision clearly does not guarantee the precision -- the accuracy or truth -- of applied mathematical statements, which need to conform with reality.
There are quite subtle ways of falling into this trap of confusing the two meanings. A believer in epicycles would likely have thought that it must have been correct because it gave mathematically correct answers. And it actually did . Epicycles actually did precisely calculate the positions of the stars and planets (not absolutely perfectly, but in principle the theory could have been adjusted to give perfectly precise results). If the mathematics was right, how could it be wrong?
But what the theory was actually calcualting was not the movement of galactic clockwork machinery and stars and planets embedded within it, but the movement of points of light (corresponding to the real stars and planets) as those points of light moved across the sky. Those positions were right but they had it conceptualised all wrong.
Which begs the question of whether it really matters if the conceptualisation is wrong, as long as the numbers are right? Isn’t instrumental correctness all that really matters? We might think so, but this is not true. How would Pluto’s existence been predicted under an epicycles conceptualisation? How would we have thought about space travel under such a conceptualisation?
The moral is, when we're looking at mathematical statements, numbers are representations, and representations can lie.
If you're interested in knowing more about epicycles and how that theory was overthrown by the Copernican one, Thomas Kuhn's quite readable The Copernican Revolution is an excellent resource.
= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
= f037147d6e6c911a85753b9abdedda8d){kind=logo}
Yes. Too often people treat it as a sin to criticize without suggesting an alternative. (as if a movie critic could only criticize an element of a film if they were to write a better film).
But coming up with alternatives can be hard, and having clear criticisms of current approaches can be an important step towards a better solution. It might take years of building up various criticisms -- and really coming to understand the problem -- before you are ready to build an alternative.