His version is idiosyncratic though. — Count Timothy von Icarus

Funny, because that's the exact word I was going to use :lol:

Here's a Proofs and Refutations - the source of Lakatos' concept of lemma incorporation - inspired investigation into square circles. — fdrake

Nice. That's the sort of playfulness we get by adopting these considerations. I can't help you with re-defining smoothness for Taxicab space, but since every point is on a corner I don't see how the path can be differentiable, and hence smooth.

This Interactive Mathematics page shows the problem, under "An interesting question arrises". There are two values for the limit - 2 and √2. So the space is not smooth, unless we re-define "smooth".

...logical impossibility isn't all it's cracked up to be... — fdrake

That's the take-away. It's related to what I was trying to show with Banno's game - in which any rule can be undermined; but also, and yet again, to the analysis of language in A nice derangement of epitaphs.

Thanks.

The chart shows that the expression you are using is not a commonplace. That's all.

Seems to me that it remains unclear what "material logic" is. But that is not true of formal logic.

A tale. One of the pre- socratics - I forget which - "proved" that air becomes colder under pressure by blowing on his figure. The breath feels cold. And we all know that a wind is cold. Hence, he disproved that gases under pressure increases in temperature. Do we take this as a refutation of thermodynamics?

Seems to me that you are truing to do something similar with formal logic. It just doesn't work. So:

Basically I see the appeal of Aristotle and common sense as a mistaken appeal — Moliere

Put bluntly, I do not see that you have differentiated formal and material logic in a way that can be maintained beyond "material logic is an over-simplification of formal logic".

And in any case, this does not address Russell's case.

I'll take your word for that. The difference here is the rigour introduced by formality. In particular, the argument is not subjective, nor deriving from perception. But yes, there are similarities to less "analytic" philosophies.

Material logic — Count Timothy von Icarus

Is this what you mean by material logic?

Historically logic is the thing by which (discursive) knowledge is produced. When I combine two or more pieces of knowledge to arrive at new knowledge I am by definition utilizing logic. — Leontiskos

Or we could say that logic is that by which correct inference is achieved.

word searches are neither good arguments nor good ways of informing yourself about philosophy. — Count Timothy von Icarus

Indeed! It is also a symptom of conceiving everything in terms of technicalities, technical terms, and stipulations.

A tale. One of the pre- socratics - I forget which - "proved" that air becomes colder under pressure by blowing on his figure. The breath feels cold. And we all know that a wind is cold. Hence, he disproved that gases under pressure increases in temperature. Do we take this as a refutation of thermodynamics? — Banno

Exactly. We don't use logic to tell us what's in the world. If we did, we'd still be in the stone age.

logical impossibility isn't all it's cracked up to be — fdrake

Well, your post would appear obtuse to the layman, and maybe it just is. Maybe the argument is much simpler than you are making it:

• Circles are round
• Squares are pointy
• What is round is not pointy
• Therefore circles are not square

Or even simpler:

• Circles are circular
• Squares are square
• What is circular is not square
• Therefore circles are not square

These arguments are not any less powerful for their simplicity, and most objections would be little more than quibbles. For example, someone might offer the counterargument of a shape like 'D', and claim that it is both circular and square. That quibble of course could be addressed, but need not be.

More formal:

• The points of a circle are all equidistant from some point
• There is no point from which the points of a square are all equidistant
• Therefore no circle is a square, and no square is a circle

It is very odd to question such arguments. If these are not good arguments, then there probably is no such thing as a good argument. There seems to be a point at which trying to be charitable towards a dubious thesis crosses over into sophistry, no? Logicians have a difficult time saying that some claim or argument is false or unsound, as opposed to merely invalid. In these cases one must recognize that falsity can enter into a concept; that someone can simply fail to understand what a circle or square is.

"Pure logic" as you describe it could never get off the ground because it would be the study of an infinite multitude of systems with absolutely no grounds for organizing said study. — Count Timothy von Icarus

A quick look at the Open Logic Project will show you how logic grows, tree-like, each system depending on, but slightly different from, the others. It's already "off the ground".

Therefore it's ok to do pointless investigations. — frank

More than OK.

When we do come across a "paradox in nature", so to speak, what we do is change the way we talk about what we see. Perhaps the commonest example now is the supposed paradox of the dual nature of particles and waves. Instead of talking about particles and waves we use Schrödinger's equations and things work nicely. Similar tales can be told about heliocentrism and the speed of light in a vacuum and many other adjustments to our understanding. We don't come across "paradoxes in nature", not because the world is made so as to avoid paradoxes, but becasue we change the way we describe things in order to accomodate what was previously spoken of paradoxically.

That is, we adapt our logic to match what we see.

:up:

Basically I see the appeal of Aristotle and common sense as a mistaken appeal -- it makes sense of the world, but need not hold for all empirical cases: There are times when a person is in contradiction with themself, or an organism has a contradictory cancer, or a social organism is composed of two opposite poles (hence Hegel's use of contradiction in attempting to understand a social body or mind). — Moliere

But these are so far from counterexamples to Aristotle that they are all things he explicitly takes up.

And I, for one, take up the liar's paradox as a good example of an undeniable dialetheia: A true contradiction. — Moliere

Every time I have seen someone try to defend a claim like this they fall apart very quickly. The "Liar's paradox" seems to me exceptionally silly as a putative case for a standing contradiction. For example, the pages of <this thread> where I was posting showed most everyone in agreement that there are deep problems with the idea that the "Liar's paradox" demonstrates some kind of standing contradiction.

Cognitive biases are odd things.

And I, for one, take up the liar's paradox as a good example of an undeniable dialetheia: A true contradiction. — Moliere

A good example of how re-thinking how we phrase the apparent paradox can provide new insight. We have "This sentence is false". It seems we must assign either "true" or "false" to the Liar – with all sorts of amusing consequences.

Here is a branch on this tree. We might decide that instead of only "true" or "false" we could assign some third value to the Liar - "neither true nor false" or "buggered if I know" or some such. And we can develop paraconsitent logic.

Here's another branch. We might recognise that the Liar is about itself, and notice that this is also true of similar paradoxes - Russell's, in particular. We can avoid these sentences by introducing ways of avoiding having sentences talk about themselves. This leads to set theory, for Russell's paradox, and to Kripke's theory of truth, for the Liar.

Again, we change the way we talk about the paradox, and the results are interesting.

And again, rejecting an apparent rule leads to innovation.

Knowing something about logic and the context helps to understand why the liar paradox is of interest. — TonesInDeepFreeze

Well, your post would appear obtuse to the layman, and maybe it just is. — Leontiskos

It is obtuse, but I don't think it just is.

A metric is a way of assigning distances to pairs of points. When you consider a space, it has a metric. The usual distance people think of is called the Euclidean distance, and it's the one you're thinking of and measure with a ruler on a piece of paper.

The thing is that the choice of metric is just that, a choice, and you can write down various other spaces with various other metrics. One of those other metrics is called the taxicab metric. Contrasting that to the Euclidean metric:

Imagine you start at a point, and you go 1 step north and 1 step northeast
The taxicab metric says you've travelled 2 total units - you add the steps.
The euclidean metric says you've travelled sqrt(2) total units - you measure the line.

Because a metric defines the concept of an interpoint distance, circles in taxicab geometry are different from circles in euclidean geometry. A circle in taxicab geometry, a set of points defined as equidistant from a single point, looks a lot like a square in euclidean space. 4 corners, 4 right angles, 4 equal sides.

So it is a circle, if a circle is defined by the property of being equidistant from a point. But perhaps it is not a circle, because... well, like you, you could insist that we're not talking about a circle when we're talking about sets equidistant from a point in the taxicab metric. So for you, you'd have to do something to block what we're talking about as a circle in taxicab geometry being a "real" circle.

That places a burden on you to study the concepts of circularity and square-iness, and to say why the first blocks the latter and vice versa. Which is what I did in the post. I'll go through it for nonmathematicians.

For something to count as a square, it needs to have:

S 1) Four sides of equal length.
S 2) Each side meets exactly two other sides at right angles.

Let's just take that as a given, that is what a square is. Now we need to think about a circle. What's a circle?

C 1 ) A circle is a set of points equidistant from one point.

If ( C 1 ) is the only defining property, the taxicab circle is indeed a circle, it's just a circle in taxicab space. Clearly you don't want it to be a circle, so you need to stipulate a restriction. I could also insist that it is a circle, and how are we to decide between your preference and my preference? Anyway, onwards:

What is round is not pointy — Leontiskos

You specified such a restriction with "what is round is not pointy", which is something similar to what I formalised with the idea of smoothness. The "corners" form the "pointy bits" of the square because the function that defines a square is not smooth at the exact corner point.

There is an ambiguity regarding pointiness, which is similar to the above ambiguity regarding equidistance. In thinking about the corners of the square thing (the taxicab circle) in taxicab space as pointy in the above sense, that requires specifying the roundness concept in terms of the measure of size - smoothness is typically characterised with respect to a measure of size.

Something is differentiable when its derivative exists at every point.
The derivative of a curve exists at a point if and only if at that point the limit of the ratio of the function evaluated at the endpoints of an arbitrarily small interval divided by the length of that interval exists (IE it becomes just a number).
A curve is smooth if you can apply the procedure above to it arbitrarily many times.

The concepts of "interval" and "length" there are also doing a lot of work, since they're distance and size flavoured. And should we expect them to work as our prior Euclidean flavour intuitions would in taxicab geometry? What gives us the right to insist that we think of smoothness as we would in a Euclidean space and transfer it onto smoothness in a taxicab space?

Clearly you would want to insist that they do, my intuitions also run that way. But my intuitions can also side with circles not necessarily being smooth since I'm used to dealing with this stuff!

Where we can agree, though, is with lemma incorporation. In which we specify a set of properties that say exactly what counts as a circle (in your sense) and why it can't be a square.

So for you:

A circle is, by definition, a set of points Euclidean equidistant from one central point.

And thus we've revealed what sneaky hidden presumption you had through lemma incorporation. What we haven't done is decided why that must be accepted as the definition of a circle.

If you want to join in with this exercise of lemma incorporation, I invite you to stipulate a definition of pointy! And we will see where it goes.

A circle is, by definition, a set of points Euclidean equidistant from one central point. — fdrake

@Leontiskos

As an aside, here are some possible counterexamples.

Take all the points Euclidean distance 1 from the point (0,0) in the Euclidean plane. Then delete the point (0,0) from the plane. Is that set still a circle? Looks like it, but they're no longer equidistant from a point in the space. Since the point they were equidistant from has been deleted.

Another one. Take the circle with radius 5 centred at the point (0,0). Then remove all points in the space which have coordinates which are both natural numbers - like (1,2), (7,8). Removing all those points removes the point (3,4), which lays upon that circle (since 3^2+4^2=5^2). That doesn't do anything to change the smoothness of the circle either, since every point on it is the same as before. So it's still smooth, no corners, all points equidistance... It's just missing a point. So, all points in that space which are Euclidean distance 5 from the origin are in the set - so is it a circle?

These would mean you have to come up with some constraint on how hole filled the space, or the circle, could be, and think about holiness itself in order to restore the fact both are clearly circles... Or maybe they're not circle at all at this point. Or neither of them are real counterexamples - it could be my specification's shite.

See what I mean?

Nice post.

...you could insist that we're not talking about a circle when we're talking about sets equidistant from a point in the taxicab metric. — fdrake

Importantly, doing this would not be wrong, as such. It's just one approach amongst many. The error here, if there is on, would be to presume that this was the only, or the correct, approach - that it's what we ought do.

Importantly, doing this would not be wrong, as such. It's just one approach amongst many. — Banno

Indeed.

Mathematics papers absolutely call taxicab-circles circles. I just wouldn't call them circles to my students learning shapes.

"An interesting question arrises". There are two values for the limit - 2 and √2. So the space is not smooth, unless we re-define "smooth". — Banno

We had a related discussion here.

My explanation for the weirdness of the staircase paradox. The tl;dr of it is that the length you get by placing a measuring tape along a curve doesn't respect the process of infinitely refining shapes. So it's nothing to do with the shape, it's to do with the concepts of length and limit.

I honestly don't have the maths to try to think about volume and rate concepts in taxicab geometry. Other than my intuition that they're the same as the Euclidean ones... even though the length is different.

Me going through the maths there isn't an attempt to side with ↪Cheshire over @Banno, because being able to explore the conceptual content of the allegedly logically impossible should tell you that logical impossibility isn't all it's cracked up to be. You do have to ask "which logic and system?", and "what concept am I not formalising right?" or "what concept is making the weird shit I'm imagining weird?".

Fair enough. But is our preference for systems arbitrary? It seems very easy to have a system where "circle" can be "square." You can even make it axiomatic.

If the presupposition is that all systems are equal, our preferences for them arbitrary, then of course logical impossibility is pretty much meaningless.

But we don't pick systems arbitrarily. It's not the case that the Earth, baseballs, and basketballs are all just as triangular as they are spherical just because it is possible to define a system where this is so. To affirm that would be to default on the idea that any statement about the world having priority over any.

I agreed with

Trick question. As long as you are talking about tiny triangles the sides add up to more than the diagonal. No matter how small they get. So the only question is what do the sides add up to in one tiny triangle. Then multiply by the number of triangles to get 2. A triangle is not a diagonal! — EnPassant

But that's cheating, of course. "Monster barring" in Russell's terms.

Seems to me that it remains unclear what "material logic" is

What about the summary here is unclear? https://thephilosophyforum.com/discussion/comment/939603

There is indeed debate over what the proper object of study is here, sure. That's also true of mathematics though. To quote Andrew D. Irvine:

One of the most striking features of [modern] mathematics is the fact that we are much more certain about what mathematical knowledge we have than about what mathematical knowledge is knowledge of. Mathematical knowledge is generally accepted to be more certain than any other branch of knowledge; but unlike other scientific disciplines, the subject matter of mathematics remains controversial. In the sciences we may not be sure our theories are correct, but at least we know what it is we are studying.”

But is our preference for systems arbitrary? — Count Timothy von Icarus

I don't think it is.

But we don't pick systems arbitrarily. — Count Timothy von Icarus

I agree. They are picked to reflect, capture or illustrate certain ideas. If you came up with a system of arithmetic that couldn't prove 1+1=2, it'd be a shitty system of arithmetic.

It's not the case that the Earth, baseballs, and basketballs are all just as triangular as they are spherical just because it is possible to define a system where this is so.

I agree. The everyday conceptual content of Earth (the concept), baseballs (the concept) and basketballs (the concept) are that they are round.

To affirm that would be to default on the idea that any statement about the world having priority over any.

I disagree. I think you missed the case that priority can also be seen as purpose and context relative. Here's a series of examples regarding roundness and sphericality.

I prioritise the notion of roundness when considering the Earth on an everyday basis, and I might while calculating its surface area - fuckit it's a sphere and that'll do. But on a day to day basis, my body treats the Earth by and large as flat. And that has priority over a merely intellectual commitment to its roundness as far as my feet are concerned. If I'm trying to stand on a bosu ball, now that fucker is round.

If I were studying variations in the acceleration due to gravity on the Earth's surface, I couldn't treat the Earth as a sphere - since it's roughly oblate, it's a spheroid. And crap like Mt Everest sticks out of it, so it's pointy. If we go by @Leontiskos intuition that round things cannot be pointy in any context, well the Earth is in trouble.

More generically, the role specifying a system has might be thought of as setting out some concept for some purpose. That allows you to see whether the system specification is fit for task.

How do you decide whether it's fit for task? Well I suppose you decide on a task by task basis. Thinking of Earth strictly as a sphere, with the assumption that a sphere is like a circle where every point on its surface is equidistant from its centre... That doesn't work as soon as your legs move. So that's not fit for walking.

But it is fit for a quick and dirty calculation of volume. Or an explanation for how it attained its shape due to gravity.

Here are more abstract examples.

Those tasks are quite concrete - there are harder ones. Like how might we consider fitness for task of a concept of logic in the context of arguing with a salesperson? Their responses aren't going to follow propositional logic... So something informal is required, they're definitely trying to persuade you. Emotional appeals? Reframing? Motivational speech? We could speak of a "logic of sales" that consists of such chicanery. And it would be nuts to think of the salesperson's behaviour solely terms of syllogisms and propositions.

How might we consider the laws of addition when considered from the perspective of raindrops? Well one raindrop alongside another raindrop might be two raindrops, but it could be one larger raindrop depending upon the distance between them. So "raindrop addition" might be way more complicated than adding discrete units of things...

Here's what I think is the general principle.

The rough trick is the same in each case, you have some conceptual content you want to specify, you try to set out a collection of rules that specify the conceptual content, then you shit test the rules to see if you got anything wrong. Or you can maybe prove all and only the results that you want - or solve all your problems - then you've succeeded beyond your wildest dreams.

What about the summary here is unclear? — Count Timothy von Icarus

So it's bits of applied logic and ontology and model theory and metalogic. Fine.

Imagine you start at a point, and you go 1 step north and 1 step northeast
The taxicab metric says you've travelled 2 total units - you add the steps.
The euclidean metric says you've travelled sqrt(2) total units - you measure the line. — fdrake

But this isn't right. The Euclidean metric says you've traveled 2 total units. Yet the distance of a straight line between your starting point and your ending point is sqrt(2). Apparently taxicab geometry measures the distance between points differently.

A circle in taxicab geometry, a set of points defined as equidistant from a single point, looks a lot like a square in euclidean space. — fdrake

Not really. Only if the radius is a single unit. The larger the radius, the more circular it will be.

I could also insist that it is a circle, and how are we to decide between your preference and my preference? — fdrake

You're presuming that your made up "taxicab geometry" is on a par with Euclidean geometry. But it's not. What you've done is engaged in equivocation. You want to say, 'A circle is the set of points equally "distant" from a single point.' Scare quotes are required, because we both know that your artificial definition of "distance" is not the accepted definition. Similarly, 'This figure is a "circle" in taxicab geometry.' But I was talking about circles, not "circles."

The derivative of a curve... — fdrake

We could say that a circle is a figure whose roundness is perfectly consistent.* There is no part of it which is more or less round than any other. In calculus that cashes out as a derivative, but folks do not need calculus to understand circles. Calculus just provides one way of conceptualizing a circle.

A circle is, by definition, a set of points Euclidean equidistant from one central point.

And thus we've revealed what sneaky hidden presumption you had through lemma incorporation. — fdrake

Is it more "sneaky" to think that circles go hand in hand with Euclidean geometry, or to think that Euclidean geometry and taxicab geometry are on a par? Not only are they not on a par; taxicab geometry presupposes Euclidean geometry.

Take all the points Euclidean distance 1 from the point (0,0) in the Euclidean plane. Then delete the point (0,0) from the plane. Is that set still a circle? Looks like it, but they're no longer equidistant from a point in the space. Since the point they were equidistant from has been deleted. — fdrake

But they are. You have an odd assumption that points are stipulative, as if we could delete a point or as if a point could have spatial extension. The set of points is still equidistant from a point. This idea of "deleting" points mixes up reality with imagination.

If we go by Leontiskos intuition that round things cannot be pointy in any context, well the Earth is in trouble. — fdrake

I think you are falling into the exact sort of quibbling and sophistry that I warned against. The answer here is simple: the Earth is not perfectly spherical or perfectly round. A cross-section of the Earth is circular, but is not truly a circle.

I just wouldn't call them circles to my students learning shapes. — fdrake

And the reason why is very important.


* And of course also possesses roundness

I agree. The everyday conceptual content of Earth (the concept), baseballs (the concept) and basketballs (the concept) are that they are round.

And why is this? Is it not because of what those things actually are? If not, why did this become the everyday concept?

How do you decide whether it's fit for task? Well I suppose you decide on a task by task basis.

Sure. So with the "raindrop" addition example, isn't the appropriateness of the system determined by the real properties of rain drops?

I am all on board with the idea that the tools will vary with the job, but it seems to me that to explain why some tools are better for some jobs than others requires including properties of "things in the world."

Even when we speak of "concepts," it seems to me that there is plenty of evidence to support the claim that our cognitive apparatus is shaped by natural selection, and this in turn means our thinking and our preferences, relate to "how the world is."

To put it succinctly, there are causes for our preferences and what we find useful. And I would also argue that these causes cannot all be traced exclusively to our minds/concepts, that our minds and concepts themselves have prior causes.

your made up "taxicab geometry" — Leontiskos

He didn't make it up.

If the presupposition is that all systems are equal, our preferences for them arbitrary, then of course logical impossibility is pretty much meaningless.

But we don't pick systems arbitrarily. — Count Timothy von Icarus

Yep. If everything were arbitrarily stipulated, then all of the strange ideas in this thread would be gold. ...Or at least as valuable as everything else.

- I realize that someone prior to fdrake made it up.

But they are. You have an odd assumption that points are stipulative, as if we could delete a point or as if a point could have spatial extension. The set of points is still equidistant from a point. This idea of "deleting" points mixes up reality with imagination. — Leontiskos

Let's change track. You tell me exactly what you mean by a circle with an intensional definition, and we'll go with that. Then do the same for roundness and pointy!

And why is this? Is it not because of what those things actually are? — Count Timothy von Icarus

I think so, relative to tasks.

Sure. So with the "raindrop" addition example, isn't the appropriateness of the system determined by the real properties of rain drops? — Count Timothy von Icarus

Yes.

I am all on board with the idea that the tools will vary with the job, but it seems to me that to explain why some tools are better for some jobs than others requires including properties of "things in the world." — Count Timothy von Icarus

Yeah that's a hard one. I don't know if there's a hard and fast answer for systems generically! This seems to be a root level epistemological issue - what it means for a description to be adequate.

Even when we speak of "concepts," it seems to me that there is plenty of evidence to support the claim that our cognitive apparatus is shaped by natural selection, and this in turn means our thinking and our preferences, relate to "how the world is." — Count Timothy von Icarus

Indeed. Though there are lots of ways what we create can model, describe or explain stuff. Maybe even mirroring different aspects of stuff. Maybe it doesn't need to do any of these things to still be important.

Not for nothing, but a square is an approximation of a circle. A better approximation than an equilateral triangle, but not as good as a regular pentagon.

But then, who would ever consider approximating curves with straight lines? Ridiculous idea.

Let's change track. You tell me exactly what you mean by a circle with an intensional definition, and we'll go with that. Then do the same for roundness and pointy! — fdrake

I hope I'm not the only one who recognizes that you are more interested in this conversation than me. :grin:

I am fine with taking Euclid's definition:

A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre. — Circle | Wikipedia

And we can say that a square is a plane figure with four equal sides and four right angles.

Something like "roundness" I take to be a simple concept, not especially reducible to further explication. We could say that it is something like the curvature of a line.