Sometimes very active discussions about peripheral issues overwhelm a thread, so this is a permanent home for those conversations.
I’ve opened a new “Sandbox” thread as a post as the new “ignore commenter” plug-in only works on threads started as posts.
Jock:
No, chopping off nonzero digits doesn’t make a number inexact. 92.3456 and 92.34 are both exact. 1.414 and √2 are both exact. All numbers are exact. Do you really think that 1.414 has more than one value?
If you round “8.379 inches” to “8.38 inches” in order to avoid overstating the precision, that doesn’t mean that the number 8.38 is inexact. It’s just that you’re using the representation to signal something about the precision of the measurement.
Representations are distinct from numbers, and numbers are distinct from measurements. If I could tattoo one thing onto the forearms of you and Flint, that would be it.
keiths:
Neil:
That doesn’t help. You’re saying that length doesn’t exist (or at least that it may not exist). If length doesn’t exist, yet you claim that you are measuring length, then you are claiming to measure something that does not exist. That’s a problem, in my book. And if you proceed to announce that object A is longer than object B, despite the fact that neither of them possesses a length at all, then you are not making sense.
If I understand your position, it would be more accurate to say “I am executing a certain procedure, and that procedure produces a number. I am attaching a unit to that number and calling it a length, though it actually is not a length, since length isn’t actually a thing.”
keiths:
Neil:
There’s nothing remotely theistic or supernatural about my statement. I’m simply saying that if I measure the length of something repeatedly and get consistent results, but I’m really not measuring the length, then there is something else out there, apart from me, that is causing me to get those consistent results. Some feature of external reality is causing my results to be consistent. Otherwise I have no reason to expect consistency.
In terms of my own view, I would say that there is something out there: the object whose length I am measuring. And length is an objective property of that object. So when I take a measurement, I am approximating the objective length of the objective object that is out there. The object is out there, and the length is out there. Measurement is a way of accessing the reality out there. In other words, there is something out there.
Unless you think that I’m unwittingly generating the results myself, and fooling myself into thinking that they come from the outside world. Do you agree that there is an external reality, apart from us, and that we interact with that external reality when we make measurements?
As a fictionalist mathematician, I also say that numbers don’t exist. Yet I use them when doing math. I’m not seeing a problem, apart from perhaps a disagreement over the meaning of “exist”.
Are you trying to start an argument on whether there are objects or on the meaning of “objective”?
Instead of arguing about length, let’s look at force. According to Aristotle, if an apple falls from the tree, that’s just its nature. No force is required. According to Newton, there is a force of gravity which causes the apple to fall. According to Einstein, there isn’t any force, there’s just the curvature of space-time. Yet reality continues to work as it did, whether or not we say that there’s a force.
I’m saying the same about length. Reality continues to work as it did, whether or not we say that there are lengths. I see length as a concept that we use in our interactions with nature, rather than something that exists independent of us.
Erik had it right. I am taking a nominalist position. In philosophical terms, it doesn’t actually matter whether you are a nominalist, a platonist or a realist. What we say about whether length (or force, or other abstraction) exists doesn’t actually matter. What matters is how we use them.
I agree with that. Our disagreement is on whether “length” names a component of that reality.
I can envision the possibility of a different civilization that develops engineering, space travel, etc, but whose science does not use the concept of length at all. They happen to have come up with a radically different set of concepts to use in their science.
I agree. And a human user of that calculator who treated 12 and 12.0 the same would not be understanding the meaning of precision. Common sense would be involved, if measurements were being used.
So once again you are saying that any number you write down, regardless of its provenance, is ipso facto an exact number, because that’s the only kind of number there is. The precision may be incorrect, the accuracy may be wrong, the tool used to create that number may be wildly inappropriate, the consumer of that number may misinterpret your context, but dammit the number is farking EXACT.
All those other considerations apply only to sensible people. I find it difficult to keep converting from what I consider a number (an actual value, generally an approximation with an error distribution) to what you consider a number, which is an abstract theoretical symbolic concept divorced from human experience. I find this theory unnecessary.
Not just the meaning of “exist”, but also of “real” and what it means to work. On your account, numbers are useful, they work, and they don’t exist and are not real. And you refuse to see a contradiction there.
This is extreme nominalism.
Neil:
OK, the analogy with mathematical fictionalism definitely helps. I too am a mathematical fictionalist, so perhaps we can find some common ground there. As a fictionalist, I’m willing to say things like “12 is 1 less than 13” in ordinary conversation, both for convenience and for conflict-free discussions with Platonists :-), but what I really mean is more like
“if the fictional entities 12, 13, and 1 were real and had the same attributes that they have in my model, then 12 would in reality be 1 less than 13.” Is that close to how you would put it?
Extending that beyond math, I take you to be saying something like “length is a fictional property of fictional objects, and what we call length measurements are procedures we follow in which we seem to use measuring devices to approximate the value that the fictional length would have if it and the fictional object were real.” The model seems real to us, and the objects in the model appear to have an objective existence, but that’s actually an illusion. The objects in the model seem real because we are imposing our model on external reality, not because external reality intrinsically bears a strong resemblance to our model.
Is that anywhere close to your actual view?
keiths:
Neil:
Definitely not. That was simply part of my answer to your question “Why does there need to be something out there?” When I measure something, I am not generating the result from within. I am interacting with external reality, and the result of the measurement originates there, even if it gets distorted and transformed on its way through my model.
I recall a discussion with you years ago in which you seemed to deny that external reality has its own features independent of the models we impose on it. I don’t find that plausible, because I think that there have to be intrinsic features of external reality that map onto things like the lengths in our models. Otherwise there would be no reason to expect consistency in our measurements.
In practical math it indeed does not matter what your background philosophy is. But as a philosopher those are different universes.
In arithmetic you put numbers together. In geometry you figure out shapes. In philosophy you apply concepts like “real” and “exists” and it makes a truly big difference what you think exists and what does not.
What is the difference? Reality doesn’t care.
keiths:
Flint:
No, a human would be correct to treat them the same. Math is math, Flint. If it’s correct for a calculator to treat those the same, then it’s correct for a human to do so as well.
You are once again confusing numbers with measurements.
Measurements are distinct from numbers. A human who blindly treated the measurements “8.6 inches” and “8.600 inches” identically would be screwing up, but a human who treated “8.6” and “8.600” as the same number would be doing exactly what she should do, and exactly what her calculator should do, as well.
Representations are distinct from numbers, and numbers are distinct from measurements. Head on down to the tattoo shop.
Truth should care. But your reality is so small and narrow that truth does not exist. So… I get you. When truth does not exist, then there’s nothing left for reality to care about. Even if your reality presumably includes you, it does not include truth.
And how do you make the distinction clear?
I don’t own reality but, sure, my perception of it is necessarily limited by my knowledge and cognitive abilities. Is the issue there’s no place for Gods in “my” reality?
keiths:
Alan:
It’s easy. Measurements include units. “5.68 inches” is a measurement. 5.68 is a number.
Accepting the convention of what an inch represents, indeed. How long is an inch?
Lol. An inch is a unit of length, Alan. “5.68 inches” is therefore a measurement.
An inch could be five miles, or it could be a micron, and “5.68 inches” would still be a measurement.
The issue is that you think you can say things like “Reality doesn’t care” and think it makes a point, when at the same time the reality you know is appallingly limited. Your reality is far more limited than your knowledge and cognitive abilities. Based on your knowledge and cognitive abilities it should be self-evident to you that e.g. truth matters, but you specifically choose to lock this out of what you call reality.
If you were consistent in your world view, you would understand that a statement like “Reality doesn’t care” is far too big for you to say. All your world view permits you to say and think are things like, “I woke up. I’m getting out of bed and have breakfast. Maybe watch a movie later. Oops I dropped my coffee.” Nothing beyond this. So the issue is that your world view is wildly inconsistent and incoherent.
The best relationship you can have with truth is like Neil: It’s a fiction. You appreciate truth like you appreciate Tom and Jerry. Or perhaps a good Sherlock Holmes story. Which is ludicrous enough, but this is the best nominalists and fictionalists can have, if they were consistent.
Well, let’s call it imagination. Though science fiction works continue to demonstrate the limits of human imagination.
Sure, but such measurements don’t communicate information without the convention of what the unit represents. Of course, when constructing your desert island shelter, you can use whatever convention is most convenient, most usefully relating to your own body dimensions, I would think.
Flint:
Yes. You think that’s a problem, but it isn’t. Your intuition is wrong. I’ve shown you this, but you refuse to engage my argument.
No, the precision cannot be wrong. All numbers are infinitely precise, and that is the correct precision for them to have. A measurement can be expressed with the wrong precision, but a number cannot. Numbers are distinct from measurements.
No. A number on its own cannot be accurate or inaccurate. It just is. Is 29.48 an inaccurate number? You can’t answer, because accuracy doesn’t apply to numbers on their own. It applies to measurements, but not to numbers.
Numbers are exact regardless of how they are created, and the fact that a number is exact doesn’t mean that it can’t be misapplied. Suppose I need to measure an object. Instead of using a measuring device, I use a random number generator. The number that comes up is 8.201, so I write down “8.201 inches”. That’s wildly inappropriate. The measurement is almost certainly wrong. And the number 8.201 is exact. It’s equal to 8.201 and to no other number.
Why should the consumer’s mistake affect the exactness of the number? If it’s exact, it’s exact.
Yes, all numbers are “farking EXACT”. Your intuition says no, and your intuition is wrong. I’ve shown you this, and you keep running away from my argument. It’s pitiful.
You don’t have to convert. The number starts out in the right form. If the Meas-o-matic reads “4.958 in”, then write down the exact number 4.958 followed by the word “inches.” Then you’re all set. 4.958 is an exact number, “4.958 inches” is an inexact measurement, and all is well with the world. Nothing goes wrong.
The irony is that it’s the MDRs that are unnecessary. They create extra work but confer no benefits whatsoever. You can’t even tell when you’re looking at one. Is 38.7 an MDR? You have no idea. The MDR 38.7 looks identical to the real number (aka IPR) 38.7. They carry exactly the same information. To “convert” the IPR 38.7 to the MDR 38.7 consists of crossing out the “38.7” and writing “38.7”:
38.738.7On what planet does crossing out a number and rewriting it count as an improvement? The MDRs are a complete waste of time.
Context, keiths. Remember the context.
Alan:
So? You asked how to distinguish numbers from measurements, and I told you. Measurements include units, but numbers do not. If you see a unit, you know you are not looking at a simple number.
Let me introduce you to a new unit of length, the fligbart. This object sitting next to me, which you cannot see, is 4.902 fligbarts long. Is “4.902 fligbarts” a measurement? Yes.
You don’t need to know how long a fligbart is in order to recognize that “4.902 fligbarts” is a measurement.
keiths, to Flint:
Alan:
Haha. “On their own” sailed right over your head, didn’t it?
See you tomorrow, Alan.
I’m more inclined to think of “exist” as context dependent. When I am doing math, I treat numbers as existing. And I might say that a number exists which solves an equation, even when I don’t know what that number is.
But when I am not doing mathematics, but perhaps talking about mathematics to a non-mathematician, then taking the position that numbers do not exist helps explain why mathematics is not an empirical science.
Have you considered the possibility that a different civilization might use very different model with radically different objects and properties, yet still successfully navigate the world. Perhaps what you take to be intrinsic features of external reality are, in part, intrinsic features of our own biology.
They actually operate on representations of numbers. They cannot operate on numbers, because numbers don’t actually exist.
I would be more inclined to say:
Measurements exist; numbers don’t exist.
I think I see the problem here. Changing the value of a number cannot possibly make it inexact, because every number is exact. What it does, is make the number incorrect, inaccurate, wrong. But always exact, no question about it. It’s up to the person reading that exact number to convert “infinitely precise” to “close enough or not close enough.” That person needs to understand that while 12 and 12.0 have the same “underlying referent”, nonetheless “more or less 12” could in real life represent a range even though the number used to represent that range is infinitely exact. Common sense might lead one to be concerned about the provenance of that number – where did it come from, who came up with it and how.
Flint:
Correct.
No, Flint. Consult the tattoo on your forearm. Numbers are distinct from measurements. Numbers by themselves cannot be right or wrong, accurate or inaccurate. If I ask you whether the number 13.74 is right, you won’t be able to answer. A number cannot intrinsically be right or wrong, accurate or inaccurate; it just is.
Measurements are a different story. “5.68 inches” can be an inaccurate measurement of something that is 19 inches long, but it could be an accurate measurement of something that is 5.67 inches long.
This principle holds true even in a pure math context. 13.74 is a correct solution of the equation x = 14.74 -1, but it is an incorrect solution of the equation x = √666. The number itself isn’t right or wrong, but it can be right or wrong as the solution to either of those equations.
That’s right. Numbers are inherently exact.
No, a person never needs to convert a number from “infinitely precise” to anything else. And it’s not possible, even if they want to. Numbers are invariably infinitely precise. 13.74 is exactly equal to 13.74, but unequal to any other number.
“5.68 inches” can represent a range of possible true values, but “5.68” cannot. The latter refers to one and only one value.
Common sense should lead you to ask those questions about the measurement “5.68 inches”, but if it leads you to question the exactness of the number 5.68, then your common sense isn’t sensible.
Actually, common sense should lead you to ask whether the number is or is not a measurement. Hopefully we agree that:
1) Measurements are by their nature approximations with error ranges.
2) Precision can be important in recording measurements.
3) Calculations with one or more measurements as inputs are no more precise than the input measurement with the least precision. Numbers beyond that level of precision are fiction. If your measurement is accurate to one part in 10, and your TI calculator gives you nine digits of precision, those extra digits are not sensible.
4) In the real outside world, most numbers are (or involve) measurements.
Flint:
For the millionth time, numbers are distinct from measurements. 5.68 is a number, “5.68 inches” is a measurement. Numbers by themselves cannot be measurements.
It’s time to take stock, Flint. For over six months, I’ve been presenting you with an argument showing that you are wrong, most recently here and here. In that entire time, you haven’t been able to point to a single error in it. You (and Jock) haven’t even responded to it, despite being asked repeatedly to do so. That’s a dead giveaway that you don’t have a refutation.
You are staring at a proof that you are wrong, and you can find nothing wrong with that proof. In your circumstances, the rational response would be to accept that you are wrong and change your position.* The only reason you cannot do this is because your intuition tells you that no, you must be correct. Your intuition is wrong. Intuitions, even strong ones, can be mistaken.
You are elevating intuition over reason. You are being deeply irrational, like many of the religious believers upon whom you heap scorn. I hate to say it, but you are basically being colewd on this issue.
Do you accept that intuitions, even strong ones, can be wrong? Reason should trump intuition. Instead, you are convinced that your intuition can’t possibly be wrong in this case.
What’s the point of this discussion if you’re simply going to assert your intuition again and again, ignoring reason?
*Or at the very least, to accept provisionally that you are wrong, but then set about trying to identify a flaw in my argument.
keiths:
Neil:
Yes, and I think that any such model, if it were to be workable, would need to be isomorphic to ours in important respects. In particular, I think that such a model would have to include a (mathematically) three-dimensional component corresponding to our 3D concept of space. In such a component, there would be something analogous to what we call ‘length’.
Our repeated measurements of the length of an object are consistent because we are interrogating some stable feature of external reality when we take those measurements. It would be the same for the alien civilization, when they are measuring the analogue of what we call ‘length’.
keiths:
Neil:
In the English language (and in most other languages, I suspect) the word “number” has more than one meaning. It can refer to an abstract quantity, but it can also refer to a representation of that abstract quantity. If I ask you to write down the number 7.49, you will — assuming you aren’t being contrary — write down the numeral ‘7’, followed by a period, followed by the numeral ‘4’, followed by the numeral ‘9’. You haven’t made an error in so doing. What you wrote down was in fact a number, because it was a representation of an underlying quantity, and such representations qualify as “numbers” in the English language.
You are correct that calculators operate on representations, but since the word ‘number’ can refer to a representation, my statement is correct:
We, too, operate on representations, even when we’re doing math in our heads. The representations in that case are present in our brains.
You and I are fictionalists, and if abstract numbers (in the sense of quantities, not representations) don’t actually exist, then it is impossible to operate on them. We have to operate on representations, and that is true whether we are doing math on paper, in our heads, on a computer, or on a calculator.
keiths:
Neil:
See above. Numbers do exist when we are using the word “numbers” to refer to representations. Measurements are constructed out of numbers and units. The measurement “5.68 inches” is constructed of the number 5.68 and the unit “inches”. If numbers didn’t exist, it would be impossible to construct such a measurement.
OK, then let’s rephrase that. It’s common sense to ask whether a representation that looks like a number is actually a measurement and not a number at all, even though it is treated as one.
The irony here is rich and delicious. Yes, intuitions can be wrong – EXCEPT yours, of course. YOU have “proof” that your interpretation is never wrong, and Jock and I can spend months trying to get through to you that ordinary people dealing with ordinary representations understand that these are approximations with error ranges. People call these numbers, consider them numbers, treat them as numbers, work with them using the properties of numbers, all the while realizing their limitations.
But hey, let’s all agree that the other person is an ineducable idiot and let it go at that. YOU will never grasp what we’re saying, that’s for sure. You are so goddam sure you are right that communication is ruled out. You don’t realize that your intuition has you barking up the wrong tree. I can only laugh at you, pity you, and be thankful I can think.
YES! And as I said, it’s common sense to ask which meaning is being used, because the word can be interpreted in different ways. I said exactly what you just said, but in your zeal to be right and superior and win, you didn’t even realize it! You know, this could suggest a cognitive issue…
Flint:
Neither a number in the sense of an abstract quantity, nor a number in the sense of a representation of an abstract quantity, can be a measurement. If I ask what “2.65” is a measurement of — length, weight, luminosity, etc. — you won’t be able to tell me. If I ask you what the units are, you won’t be able to tell me. If I ask you whether it is a measurement in the first place, you won’t be able to tell me, because according to you, you would need to know its provenance in order to make that determination. “2.65” behaves like a number, not like a measurement, and that’s because it is a number, not a measurement.
That made me smile. You’re mocking yourself when you say things like that, Flint. Whether you will ever recognize it is an open question.
Flint:
No. Neither a number-as-quantity nor a number-as-representation can be a measurement. It’s just a number. What is “5.31” a measurement of? Nothing. There are no units. “5.31 inches” can be a measurement, but “5.31” cannot.
Of course my intuitions can be wrong. That’s why I’m not relying on them. Instead, I’m presenting an argument.
I have an argument showing that my claim is correct. Disagree? Then refute it. It’s been over six months. It’s time for you to be brave and actually address it.
Not so. Ask someone to add 1.37 and 2.42. They’ll tell you 3.79, unless they’re really bad at math. They won’t say “I don’t know. Those numbers are inexact, so I can’t tell you what their sum is.” As you and Jock acknowledge, even the simplest statements involving MDRs can’t be answered. I wrote earlier:
What’s even funnier is that each of the first two statements has literally a zero probability of being true (since those MDRs have infinitely many values).
I don’t think you’re an ineducable idiot. I would’ve given up by now if I thought that. As I wrote earlier:
Flint:
I completely grasp what you’re saying, I recognize the error you are making, and I even understand the intuition behind your error.
The fact that we disagree doesn’t mean that we aren’t communicating. We are. The problem lies elsewhere, in your refusal to address the argument I have repeatedly put forth.
If I am wrong, there must be a flaw, or flaws, in my argument. Quote the part you think is wrong and explain why. Refute it, in other words. And since we both know you can’t do that — you would have done so long before now if you actually could — then you need to ask yourself whether you have the will to be rational, or whether you consider your intuition too strong to be doubted, even in the face of an argument you cannot refute.
Your intuition is wrong, Flint.
Lol.
In the real world, nearly every number we encounter is either implicitly or explicitly associated with a unit of some sort. And nearly always, we can determine from context what units are implied if they are not stated explicitly. The exceptions generally are NOT between units and abstract context-free numbers, but between various measurement systems – imperial vs. metric, celsius vs. Fahrenheit, etc. You are quite correct that if you make the effort to guarantee that the context is unspecified and unknown, then I can’t tell you if it’s a measurement. In real life, we have contexts.
Chuckle. You sounded like Trump, accusing every else of what he himself was doing. In the process, you mocked yourself mercilessly.
The flaw is, Jock and I are generally talking about measurements, or about values whose representation must be approximate (irrationals), and you insist on changing the subject. If you wish to believe that numbers have the magical property of being infinitely precise in principle in all cases where no units are associated, I don’t think anyone is disagreeing with you. The discussion reads like:
Flint: all measurements are approximations with ranges
keiths: numbers have no units, and are always perfect.
Flint: In real life, measurements are important and recognizing that they aren’t exact is often critical.
keiths: numbers have no units, and are always perfect.
Flint: In practice, both precision and accuracy are important.
keiths: numbers have no units, and are always perfect.
Flint: By convention, the number of significant digits represented implies measurement precision.
keiths: numbers have no units, and are always perfect.
Flint: the represented value of a measurement is nearly always a sample value taken from the error range, but there are outliers.
keiths: show me where my argument is wrong!
This long discussion about numbers, precision, etc, started in another thread. And perhaps one of my comments initiated it. But it has mostly been a ridiculous argument, and I have mostly stayed out of it because it has been so absurd.
I did remark, in the earlier thread, that it looks like a game of Calvin Ball.
Consistent with Calvin Ball, here we have Calvin (i.e. keiths) changing the rules so that he can declare victory.
I’ll note that the distinction between numbers and representations was an important part of what keiths was arguing in the earlier thread.
And here we see Calvin (i.e. keiths) claiming victory.
Ceci n’est pas une pipe.
petrushka:
Good example!
For readers who aren’t familiar with that phrase, it translates as “This is not a pipe” and it comes from the Magritte painting below.
The actual pipe is distinct from its painted representation in the same way that numbers are distinct from their representations.
Flint,
At various points in the thread, you’ve come to accept
1) that numbers are distinct from their representations, and
2) that measurements are distinct from the numbers used to express them.
Somehow, you manage to keep unlearning those. What’s up with that?
It’s why I keep joking about how you should tattoo them on your forearm.
Like Neil, I grow tired of playing Calvinball. But as a reminder to tattoo on YOUR forearm, let me say:
“since the word ‘number’ can refer to a representation, my statement is correct”
Yep, there we have it. If it’s an infinitely accurate value and ONLY an infinitely absolute value, you’re right. EXCEPT when it refers to a representation, in which case you are ALSO correct.
Heads you win, tails you win. Neat gig. And you want me to bet against you? You have defined yourself as the winner before we even start, even if winning means you get to change the rules, change the meanings of words, and insist on the opposite of what you insisted last time. Sorry, play with yourself – you can’t lose!
Flint, I’ve bolded some words in your comment for emphasis.
Flint:
Far from changing the subject, I address measurements head on in my argument (here and here, for your convenience.) But I think you already knew that, didn’t you?
Yes! What you say is absolutely true of measurements.
Yes! It’s crucial to remember that measurements are inexact and that measurement error needs to be carefully considered and handled.
Yes! The precision and accuracy of measurements are important.
Sometimes this is true, other times it is not (don’t forget my “six feet tall” example). But we are talking about the precision of the measurement, not of the number used to express that measurement.
Measurements are always inexact, meaning that the measured value differs from the true value of the thing being measured.
Please, please, get that tattoo. Measurements are distinct from the numbers used to express them.
“Isomorphic” means the same shape. But shape is itself a metric property. Until you have metrics (such as length) you cannot even talk of shape. So, to me, your argument seems circular.
Dimension is, in some sense, a metric property. Yes, we can define dimension in terms of topology without first needing a concept of length (or distance). However, topology still derives from geometry so still has metric aspects to it.
Growing up, I noticed that there were two incommensurable ways of measuring distance. One of those use miles or metres or similar so was based on a yardstick. But the other used nautical miles and was based on the angle subtended at the center of the earth. The nautical miles way of measuring distance was better suited to navigation.
Yes, those have now been merged, with nautical mile now also defined in terms of yardsticks. It was probably because of the availability of GPS satellites, that they could be merged without causing problems for navigation.
That there were, in the past, these two incommensurable ways of measuring distance argues against the idea that distance comes from nature.
I did suggest that how we see the world might be, at least in part, a feature of our biology rather than a feature of the world. Because our bones are fairly rigid, in effect we each carry portable yardsticks with us. And this may be an important part of why we measure distance as we do. Perhaps an octopus has different ways of looking at the world.
Flint,
Regarding this comment, are you telling me that you, a native speaker of English, are unaware that the word ‘number’ has more than one meaning? That it can refer to a value as well as to a representation of that value? I’m not buying it.
If someone asks you to write down the number 4.5, do you respond “I can’t. You can’t write down a number, only its representation”? Please.
keiths:
Neil:
Yes. In a dastardly move, I ran out and changed every English dictionary in the world, right in the midst of our conversation. How shameful.
Amen. And it’s obvious that I was using ‘number’ to refer to quantities there, and not representations. Otherwise, I would have been saying “representations are distinct from their representations.”
keiths:
Neil:
Yes, if by “claiming victory” you mean showing that my statement is correct.
Are you going to complain “It’s not fair! He’s using ‘number’ in a way that English speakers use it!”?
‘Number’ has more than one meaning, Neil.
But if you want to accost people and tell them that calculators don’t operate on numbers, be my guest. At best, you’ll get some funny looks, because people know that calculators do operate on numbers.
Neil:
The Greek roots translate as ‘same’ and ‘shape’, but that does not mean that the English word means ‘the same shape’. Here’s the Wikipedia definition:
keiths:
Neil:
By that logic, temperature does not come from nature because it can be measured both using infrared thermometers and the expansion of mercury in glass tubes, aka mercury thermometers. But that’s silly. Temperature comes from nature, and so does distance.
Distance is just as important to octopuses as it is to us, their lack of a skeleton notwithstanding. You can’t successfully hunt prey or evade predators if you are unable to perceive distance.
As a mathematical fictionalist, you ought to recognize that anything mathematicians describe as structure is dependent on abstractions and is human invented. Your response actually supports my point that you are indulging in circular reasoning.
I’m not seeing any logic at all in that response. As far as I know, there have never been two mutually incommensurable ways of determining temperature. You appear to have completely missed the point.
How many octopus witnesses can you bring to court to give witness to their use of “distance”?
If we try to describe octopus behavior, we would likely use “distance” in our description. But we cannot assume that’s how octopuses think about it.
You clearly do not understand what I am suggesting. And if you were trying to understand my thinking, you would raising different issues from what you are using. It sure looks as if you want to transform this discussion into another game of Calvin Ball. And if you continue with that, I’ll stop participating.