Are you sure we should call something like d=0.5∗g∗t2 a mathematical truth? I thought it was only true on some interpretation; as it stands, it has no meaning. — J

${d = 0.5 * g * t^{2}}$ is a mathematical equation and it works. It is true that it works, but that does not mean it is a mathematical truth.

@RussellA: Eventually, after many attempts, we invent the equation ${d = 0.5 * g * t^{2}}$, discover that it works, and keep it.................We know it works, but we don't know if it is a necessary truth.

What do you mean that the equation ${d = 0.5 * g * t^{2}}$ has no meaning?

"Are the equations being imposed or simply reflected in the mathematics?" — schopenhauer1

Both. For 100 days we observe the sun rise in the east, and invent the rule "the sun rises in the east". The rule reflects past observations, but is no guarantee that the rule will still apply in the future. We impose the rule on the world, in the expectation that the rule will still apply in the future.
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Some neo-Logos philosophies might say the mind cannot but help seeing the very patterns that shape itself. — schopenhauer1

I'm with Kant on that.
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I can imagine a type of pattern whereby the mind works (X), and a pattern whereby the world works Y, and X may be caused by Y, but X is not the same as Y. — schopenhauer1

Exactly. A postbox emits a wavelength of 700nm ( Y) which travels to the eye which we perceive as the colour red (X), where our perceiving the colour red in the mind was caused by the wavelength of 700nm in the world.

There is the general principle that an effect may be different in kind to its cause. For example, the effect of a pane of glass breaking is different in kind to its cause of being hit by a stone.
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Is our language contingently relating with the world or necessarily relating to the world. — schopenhauer1

Perhaps the difference is temporal. Going forwards in time, from cause to effect, the pane of glass of necessity breaks when hit by the stone. Going backwards in time, from effect to cause, the breaking of the glass was contingent on being hit by a stone, but equally it could have been hit by a bird.
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I can see a sort of holistic beauty in the aesthetic of the language reflecting the world because it is derived from (the patterns) of the world. — schopenhauer1

Aesthetics is perceiving a unity in the whole from a set of disparate parts. For example, the magic of a Monet derives from the artist's deliberate attempt to create a unity out of a set of spatially separate blobs of paint on a canvas. Such a unity exists only in the mind of the observer, not in the world, in that one blob of paint of the canvas has no "knowledge" as to the existence of any other blob of paint on the canvas. Patterns only exist in the mind, not the world.

As patterns don't ontology exist in the world, but do exist in the mind, to say that patterns in the mind have derived from patterns in the world is a figure of speech rather than the literal truth.
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I can see a sort of nihilistic "contingency" in the aesthetic of language never really derived from, but only loosely reflecting the world. — schopenhauer1

Perhaps it is more the case that the aesthetic brings meaning out of the meaninglessness of nihilism. It is the aesthetic that discovers the unity of a whole within disparate parts, finds patterns in randomness and seeks sense out of senselessness. For example, the aesthetic of Picasso's Guernica shows us the possibility of a greater good born out of the nihilism of war, and the aesthetic of the mathematical equation shows us a greater understanding born out of a nihilistic Universe that is fundamentally isolated in time and space.

You'd written this:

Are mathematical truths necessary truths
Following the schema "snow is white" is true IFF snow is white as a definition of "truth", then "d=0.5∗g∗t2" is true IFF d=0.5∗g∗ — RussellA

So I assumed you regarded d=0.5∗g∗t2 as a mathematical truth.

What do you mean that the equation d=0.5∗g∗t2 has no meaning? — RussellA

Without some interpretation, some assigning of the symbols, it says nothing. I can vouch for this personally, as I have no idea what d, g, and t refer to in this equation! If you simply placed it in front of me, uninterpreted, and asked me what it meant, I could only shrug.

OK, but you can see that even this weakened version represents an entirely different order of explanation than:

1. Real life effectiveness

2. Fulfills emotional desires

3. Fulfills a power structure

By referring to "accurate representation," you've introduced an epistemologically normative factor that is nowhere implied in the first three factors.

So I assumed you regarded d=0.5∗g∗t2 as a mathematical truth. — J

I would agree that the equation ${d = 0.5 * g * t^{2}}$ is a mathematical truth IFF ${d = 0.5 * g * t^{2}}$ is the case in the world.

However, who knows what is the case in the world?

I understand. Do you think there are mathematical truths that are independent of what is the case in the world? Plain old theorems, in other words?

Both. For 100 days we observe the sun rise in the east, and invent the rule "the sun rises in the east". The rule reflects past observations, but is no guarantee that the rule will still apply in the future. We impose the rule on the world, in the expectation that the rule will still apply in the future. — RussellA

Sure, but then, what of the propensity for uniformity or rules in the first place? The fact that it does act with regularities? Hume wants to skepticize this deriving or a rule as "habits of thought", but surely the habits are not like socially conventional habits like shaking hands or bowing. These are ones that nature is making, and we are taking note.

Some neo-Logos philosophies might say the mind cannot but help seeing the very patterns that shape itself.
— schopenhauer1

I'm with Kant on that. — RussellA

So you would be against the notion that the patterns are "of the world"? So, the neo-logos philosophies might say something like, "If nature has patterns, and our language has patterns, and we are derived from nature, it may be the case that our language is a necessary outcome of a more foundational logic". Thus, the logic would not be transcendental, but (for lack of a better term) "immanent" in nature, not some outside observing entity that is detached from it. There is a necessary connection proposed between noumena and phenomenal activities, but not in the "static" way of Kant, but perhaps evolutionarily conceived- there is no clear boundary as it is all derived from the same "logos".

I can imagine a type of pattern whereby the mind works (X), and a pattern whereby the world works Y, and X may be caused by Y, but X is not the same as Y.
— schopenhauer1

Exactly. A postbox emits a wavelength of 700nm ( Y) which travels to the eye which we perceive as the colour red (X), where our perceiving the colour red in the mind was caused by the wavelength of 700nm in the world.

There is the general principle that an effect may be different in kind to its cause. For example, the effect of a pane of glass breaking is different in kind to its cause of being hit by a stone. — RussellA

But my metaphor was not just of any cause, but of how language connects to reality. Neo-logos philosophies might indicate that language is structured such that it must see "reality" as it is, to be useful. It is not happenstance that language allows us to describe reality with a great degree of success. Kant never explains why our minds would compose such a world, but evolution does. Patterns of the world become sufficiently complex as to see their own patterns. Other animals are driven by the consequences of the patterns, but humans can see the causal connections, reasons, create plans, etc. All this is due to our linguo-conceptual framework our brains developed through evolutionary factors.

Aesthetics is perceiving a unity in the whole from a set of disparate parts. For example, the magic of a Monet derives from the artist's deliberate attempt to create a unity out of a set of spatially separate blobs of paint on a canvas. Such a unity exists only in the mind of the observer, not in the world, in that one blob of paint of the canvas has no "knowledge" as to the existence of any other blob of paint on the canvas. Patterns only exist in the mind, not the world.

As patterns don't ontology exist in the world, but do exist in the mind, to say that patterns in the mind have derived from patterns in the world is a figure of speech rather than the literal truth. — RussellA

This is exactly what is being questioned. Wouldn't evolution put a connection between the efficacy of the mind and the world? Prior to evolutionary theory, it was perhaps easier to detach the two and remain the ontological skeptic. Perhaps with evolutionary theory, we can think in terms of how ontology shapes epistemology.

Perhaps it is more the case that the aesthetic brings meaning out of the meaninglessness of nihilism. It is the aesthetic that discovers the unity of a whole within disparate parts, finds patterns in randomness and seeks sense out of senselessness. For example, the aesthetic of Picasso's Guernica shows us the possibility of a greater good born out of the nihilism of war, and the aesthetic of the mathematical equation shows us a greater understanding born out of a nihilistic Universe that is fundamentally isolated in time and space. — RussellA

This is true. Good observation. But what is the world outside of an observer? This goes back to the old realist/idealist debates. Is it just the case that we are simply "adding value" (in a literal and metaphorical way), or does the world already have this in itself. Think of things like "information theory", which puts information prior to the animal. But it need not be this, it just needs to be a sort of pattern that can create patterns that can understand itself. In this view, the "aesthetic" is holistic in that the observer is a natural component of the whole. In the Kantian view, however, the observer is this transcendental alien that transforms the "noumenal" into something understandable to itself. Whence this disconnect then? What to make of the two, their origins, and their connection?

I understand. Do you think there are mathematical truths that are independent of what is the case in the world? Plain old theorems, in other words? — J

Taking an example. Do I think that the theorem "angles on one side of a straight line always add to 180°" is true independent of what is the case in the world.

What is truth?

My understanding of truth is that it is defined by the schema "snow is white" is true IFF snow is white, where "truth" is the correspondence between propositions in language and equations in mathematics and what is the case in the world.

If I am correct, then if a proposition in language or an equation in mathematics is independent of what is the case in the world, then by the definition of truth, such a proposition or equation can neither be true nor false.

A proposition may work, such as "the sun rises in the east" and an equation may work, such as "1 + 1 = 2", but the fact that they work doesn't mean that they are true, if truth is defined as a correspondence between something in the mind and something in the world.

The problem is knowing what is the case in the world.

I have found the SEP article on Mathematical Explanation, which I haven't read yet, but it should make clearer Lange's idea of dividing 23 strawberries equally amongst three friends.

My understanding of truth is that it is defined by the schema "snow is white" is true IFF snow is white, where "truth" is the correspondence between propositions in language and equations in mathematics and what is the case in the world.

If I am correct, then if a proposition in language or an equation in mathematics is independent of what is the case in the world, then by the definition of truth, such a proposition or equation can neither be true nor false. — RussellA

The problem I have with these definitions is it implicitly indicates a Kantian response, but then denies epistemology proper for some deflationary "logic-only" based answer. But this cannot be the case because implicitly by saying "independent of" and "case in the world", you are using epistemological considerations, even if implicitly. These epistemological explanations require meta-logical theory, not simply refer to the correspondence or (non-correspondence) itself, but why and what and how, etc. Otherwise it's just "I have believe" without an explanation, which though is valid in terms of asserting an idea, is not necessarily valid as an fully informed reason for why you think that way. Saying "Snow is white IFF it is the case that there is at least one case of snow being white", has many implications beyond the "satisfying" of snow being white. What is "case" mean? Why are we trusting what case means? Why would if it satisfies the case you assert something like logic is "independent of" the case? etc. etc.

Unum in the same sense as in non-dualism, advaita, non divided. — Wayfarer

Or perhaps indivisible, and that seems to be a bit different. A chicken is indivisible. To divide it is to lose your chicken.

My understanding of truth is that it is defined by the schema "snow is white" is true IFF snow is white, where "truth" is the correspondence between propositions in language and equations in mathematics and what is the case in the world. — RussellA

Is this meant to be Tarski's view? Surely he didn't talk about what was the case in the world -- only about the correct relations between language and metalanguage. If one language has to be "about the world," then we wouldn't have any logical or mathematical truths at all, or at least that seems to be the necessary consequence. I don't think Tarski intended this. Unless I'm mistaken, he included these kinds of truths in his schema.

Or perhaps indivisible, and that seems to be a bit different. A chicken is indivisible. To divide it is to lose your chicken. — Leontiskos

But notice:

There was consensus among the scholastics on both the convertibility of being and unity, and on the meaning of this ‘unity’ —in all cases, it was taken to mean an entity’s intrinsic indivision or undividedness. — Being without One, by Lucas Carroll, 121-2

This is the principle that animates all living beings, from the most simple up to and including humans. it is why, for instance, all of the cells in a living body develop and differentiate so as to serve the overall purpose of the organism. So the 'one-ness' of individual beings is like a microcosmic instantiation of 'the One'. It is the basis of 'bio-logos' and the reason why Aristotelian biology retains a relevance that his physics does not

For example, the magic of a Monet derives from the artist's deliberate attempt to create a unity out of a set of spatially separate blobs of paint on a canvas. Such a unity exists only in the mind of the observer, not in the world, in that one blob of paint of the canvas has no "knowledge" as to the existence of any other blob of paint on the canvas. Patterns only exist in the mind, not the world.

As patterns don't ontology (sic) exist in the world, but do exist in the mind, to say that patterns in the mind have derived from patterns in the world is a figure of speech rather than the literal truth. — RussellA

But, all organic life displays just the kind of functional unity that a painting does, spontaneously. Those patterns most definitely inhere in the organic world. DNA, for instance.

the neo-logos philosophies might say something like, "If nature has patterns, and our language has patterns, and we are derived from nature, it may be the case that our language is a necessary outcome of a more foundational logic". — schopenhauer1

I am afraid we are not rid of God because we still have faith in grammar. — Nietszche

This is the principle that animates all living beings, from the most simple up to and including humans. it is why, for instance, all of the cells in a living body develop so as to serve the overall purpose of the organism. So the 'one-ness' of individual beings is like a microcosmic instantiation of 'the One'. — Wayfarer

Yes, and it's hard to say exactly how the animate and the inanimate relate on this score, but the convertibility of being and unum goes beyond animate realities. A rock, a molecule of H2O, a drop of water, a road, a river, a country, etc., are all one. They all possess a unitariness, so to speak, both as concept and reality.

Well, I know it's all off-topic for this thread, but that passage you quoted resonated with me.

By referring to "accurate representation," you've introduced an epistemologically normative factor that is nowhere implied in the first three factors. — J

No, that's not what I imply in my work.

There are two types of knowledges, distinctive, and applicable. Distinctive knowledge is the knowledge of identity. What we experience, and what we we distinctively experience, is known to us as is. So if I have a definition of blue or grue, I distinctively know what those terms are. The reason why we use one term over the other are what the three points cover.

At the point I try to apply those terms to reality, I have applicable knowledge. That's when I attempt to map the definition or distinctive memory that I have to reality. If I can do so deductively, and reality does not contradict me, then I applicably know that color as grue or blue.

The three points are about the question, "Which distinctive identities would a society use and deem 'the proper one'?" So if you had a group of one people who used grue, and another group that used blue, the three points I iterated above would influence which would most likely be used if the cultures were to discover each other.

Well, perhaps, but how can "accuracy" be a factor at all? What would make something an "accurate representation," to use your phrase, and of what is it a representation? None of the three factors talk about how such an idea could arise.

To put it in simple terms (borrowed from Sider), are we really not in a position to say that the Bleen people have gotten something wrong?

Maybe "it just is"? But isn't that super-convenient for us?

I mentioned this in another thread, but once you allow "it just is," for some explanations you lose the ability to exclude it for any other. If things can happen for "no reason at all," there is, by definition, no constraint on such things occurring at random.

For instance, it has been popular to say: "the Big Bang just happened, looking for reasons or caused is meaningless." Yet if we accepted this, we could never have developed the theory of cosmic inflation, now widely thought to be prior to and responsible for the Big Bang. People will say things like, "we have a sample size of one, who's to say the extreme low entropy of the universe is unlikely, it just is." Yet this same sort of argument could have been made for all the unexplainable observations that led to development of the Big Bang Theory as well. And almost certainly, if we had a good explanation for the early universe's low entropy or other elements of the Fine-Tuning Problem, it seems highly probable proponents of the "brute fact " view would abandon their position. In ways, its invocation seems very similar to the "God of the gaps."

To put it in simple terms (borrowed from Sider), are we really not in a position to say that the Bleen people have gotten something wrong?

Well, for one, color is readily phenomenologically accessible, something we experience directly, while the "creation date," of an object is not.

I feel like this is a topic where there is a pernicious tendency to prioritize potency over act. Yes, there are myriad ways in which people could categorize and conceptualize the world. We can dream up arbitrary categories all we'd like; yet the fact remains that no one actually uses them. This isn't "for no reason at all," or explicable only in terms of cultural inertia. There is a reason disparate cultures have terms for color or shape, as distinct, instead of blending some colors with shape in a single term, or some tastes with other colors, etc.

I also tend to have a somewhat Hegelian view here in that concepts simply do not evolve arbitrarily. We don't think of water as a "polar molecule" for reasons that have nothing to do with what water is. We can imagine arbitrary terms or categories; getting people to use them is another thing. The evolution of our concepts of things isn't just related to culture, it is in part determined by what those things are. And because our ends are furthered by the causal mastery that comes with techne, we are naturally oriented towards refining our understanding throughout history.

Hegel has it that social institutions "objective morality" for the individual. I'd argue that science and technology play the same role for the relevant concepts. These aren't absolute, technology evolves just as much as moral attitudes regarding slavery, but they also don't evolve at random.

We could also call on semiotics here. There is a reason why signs cause us to experience or think of anything as "that sort of thing and not any other" (the mode of causality particular to signs). Some signs are stipulated, and some are even largely arbitrary, but that doesn't negate the existence of natural signs (e.g. smoke as a sign of fire, dark clouds as a sign of rain, etc.)

Well, perhaps, but how can "accuracy" be a factor at all? What would make something an "accurate representation," to use your phrase, and of what is it a representation? None of the three factors talk about how such an idea could arise.

To put it in simple terms (borrowed from Sider), are we really not in a position to say that the Bleen people have gotten something wrong? — J

Accuracy is point 1.

1. Real life effectiveness

As long as an identity and its application are effective, or not contradicted by reality, people will hold it. Physics is held because it works. When it doesn't work, we look for an amendment or something wrong. Have you ever heard of phlogiston theory? https://en.wikipedia.org/wiki/Phlogiston_theory

At one time, it was considered a serious contender for why things would catch on fire. In short, the theory that was things which could burn had a substance called phlogiston in them that would burn when you exposed it to air. It has a few problems however, such as that some substances when burnt grew heavier, which couldn't happen if phlogiston was burning away. It was eventually replaced with Oxygen theory because it was more accurate and effective at describing the world.

If the Bleen people accurately describe a color that is useful, then who are we to care if they use the term? Maybe we don't like it, or we want our word to be dominate for some status reason, but if its a perfectly cromulent word, why not? :)

chickens — J

I think what's strange about this problem is that the setup makes human beings helpless before the implacable necessity of mathematics, and that's the wrong story to tell.

In so many cases, it is the use of mathematics that enables us to identify problems, clarify them, and solve them. And there is immense creativity here ― which is why I gestured at the invention of rational numbers.

In real life, a case like this is more likely to play out this way: you've got these 23 thingamabobs, and there's talk of splitting them three ways. You say, "Won't work," and someone less numerate than you says, "Well, let's just try." As they fail, with a puzzled look, they say, "Wait, I messed up somewhere. Let me start over." You will want to explain to them that it's impossible, because 23 is not only not a multiple of 3, it's a frickin' prime.

What's of primary interest here is that you, because of your relative expertise in mathematics, understand the situation better than the person who, even after trying and failing several times, still believes it might be possible.

But I'll address what the philosophers want to say. First, of course this scenario only makes sense given the relative durability of the thingamabobs across the sort of time scales we're interested in, and they have to be such that we can reliably distinguish them and count them. Our faculties must persist too. On and on. Absolutely there are prosaic physical qualities of the situation assumed.

All of those physical factors are also presumed in the case where we have 24 thingamabobs. In that case, if the divider-up failed, you would be the one to say, "You must have messed up, because it definitely can be done."

There's a disjunction in there right? In the first case, with 23, it was "Either I messed up, or it's impossible"; in the second, with 24, it's "Either it's impossible, or I messed up." Same thing, but with a different expectation. For the first sort of situation, it is sometimes much easier to determine that the task was impossible, than to confirm that no mistake was made in any attempt. In the second case we rely, again, on it being easier to confirm the possibility (of evenly dividing 24 by 3) than to figure out where you went wrong. Same thing again!

But because we short-circuit the disjunction differently, we're actually using it in slightly different ways. In the first case, you're discharged of responsibility for your performance because the task is impossible ― for all we know or care, you did mess up, but that's not why you failed. In the second case, we know it's not impossible, so you must have messed up; here we do judge your performance, and your mistake is why you failed. (We need a little more here actually: some guarantee that an algorithm exists, some cap on its complexity, our ability to implement it, and so on. You might still be off the hook.)

There are several options we pass by in such reasoning: we say, it's impossible, thus you needn't or shouldn't try, not that you cannot try or must not try; we say, it's possible, thus you can succeed and maybe ought to succeed, not that you must succeed.

Now come back to "why". Given 23 thingamabobs, does mathematics guarantee failure? No. It guarantees only the conditional, if you try then you will fail. (As Simpson noted, "Can't win, don't try.") Given 24 thingamabobs, does mathematics guarantee success? No, of course not, not even if you try.

So mathematics cannot compel you to succeed or to fail, but it does play a role in how we judge performance.

I think what's strange about this problem is that the setup makes human beings helpless before the implacable necessity of mathematics, and that's the wrong story to tell. — Srap Tasmaner

That seems a rather strange way to express it, but what is your alternative? "If we are smart we will foresee that the chickens cannot be evenly divided, and therefore we will not try and will not be thwarted?" Either way the math/reality constrains our options.

In real life, a case like this is more likely to play out this way: you've got these 23 thingamabobs, and there's talk of splitting them three ways. You say, "Won't work," and someone less numerate than you says, "Well, let's just try." As they fail, with a puzzled look, they say, "Wait, I messed up somewhere. Let me start over." You will want to explain to them that it's impossible, because 23 is not only not a multiple of 3, it's a frickin' prime.

What's of primary interest here is that you, because of your relative expertise in mathematics, understand the situation better than the person who, even after trying and failing several times, still believes it might be possible. — Srap Tasmaner

I think that is a large part of the point, yes. Mathematics provides us with a grasp of reality even though it is not necessary in the strict way that we tend to conceive of it. But looked at from a different angle, there is very little difference between the less numerate and the more numerate. The less numerate just takes a few more minutes to recognize that something cannot be done. And it's not as if the more numerate recognizes this a priori, in no time at all.

what is your alternative? — Leontiskos

In an artificially bounded task like this, with artificial bounds on the means by which we may complete it, there are no options. Life is not like that. This task would arise as a potential option in furtherance of another goal. Either we find a creative way to complete this subtask (making do with rough equality ― 7 or 8 each, cutting the strawberries, if that's an option, or switching measures, say from units to weight, and so on) or we mark this path off in the search and backtrack until we find a path.

Either way the math/reality constrains our options. — Leontiskos

Reality, sure, but mathematics is how we conceptualize our situation and can inform both our choice of action and our method. Mathematics is adverbial.

Anywhere you want to look, it is plain as can be that thinking and acting mathematically is empowering for humans, not some implacable constraint. Gravity is a constraint, but once we conceptualize it and understand it mathematically we put ourselves in a position to work around it, or to put it to use.

there is very little difference between the less numerate and the more numerate. The less numerate just takes a few more minutes — Leontiskos

For a toy problem like this, maybe, maybe not. I'm not sure you're entitled to this assumption.

But for the general distinction in approaches, which this little problem illustrates, the entire business world disagrees with you, the natural sciences disagree with you, the various branches of engineering disagree with you.

Well, I know it's all off-topic for this thread, but that passage you quoted resonated with me. — Wayfarer

Great. And thinking about it again, "indivisible" is probably not a great way to describe it. But it is somewhat different from advaita, at least in the sense that the undividedness is applied to being itself, which includes beings (plural). I.e. beings are also unified in their separateness. But I have not studied this question in any great detail.

-

I think @J's OP is interesting. It is something like: if mathematical necessity is not self-supporting, then whence is the necessity derived? There is an understandable temptation among some in the thread to grow impatient and fast-forward to the end instead of watching the whole movie. And that further question is a difficult one, having to do with such things as the transcendental of unum.

But the very idea that mathematical necessity is not self-supporting is important, even before we get to the further question. Mathematics seems to dominate logic, thinking, and philosophy in every age. We have a very strong intuition that mathematical necessity is necessity par excellence, and that it should be the model for thinking and reasoning. It is not obvious, historically or culturally, that mathematical necessity is not self-supporting or self-sufficient, and much could be gained by recognizing this.

In an artificially bounded task like this — Srap Tasmaner

Hmm? I find your lack of bounds more artificial than the OP. It is not artificial to say that there are unbroken wholes, such as chickens. The notion that everything is infinitely divisible is much more contrived than the alternative.

Life is not like that. — Srap Tasmaner

Sure it is. That's the point of the OP: life is exactly like that. I go to my sister's house and there are three kids who want to play with the same toys. Toys are unbroken wholes. The OP is immediately relevant. I go to the car dealership and I am offered whole cars. They don't let me buy a half car for half the money. Life is exactly like this.

Either we find a creative way to complete this subtask (making do with rough equality ― 7 or 8 each, cutting the strawberries, if that's an option, or switching measures, say from units to weight, and so on) or we mark this path off in the search and backtrack until we find a path. — Srap Tasmaner

If the OP were saying that it is a great tragedy that we can't divide the 23 chickens equally then it would be a dumb OP, but I don't see it saying that. What the OP is illustrating is equally present in all of the cases you are presenting. The curious relation between mathematics and reality is equally present with strawberries, and weight, and measurements of time, etc.

Reality, sure, but mathematics is how we conceptualize our situation and can inform both our choice of action and our method. Mathematics is adverbial. — Srap Tasmaner

But to a large extent it's not. If you think the indivisibility of the 23 chickens is merely a conceptual problem, then provide a different conceptualization in which the chickens can be equally divided. Can you do that?

You want to talk about choosing a different course, but the mathematics precedes that pivot. We choose to weigh the chickens instead of count them because we can't divide them in a numerically equal way. This decision doesn't moot the point of the OP, it presupposes it. We decide to weigh them because mathematics is not merely "how we conceptualize our situation." When my low fuel light comes on I will be able to drive for about 70 miles without fueling, regardless of how I conceptualize my situation.

Anywhere you want to look, it is plain as can be that thinking and acting mathematically is empowering for humans, not some implacable constraint. — Srap Tasmaner

It seems to be both, no? "Empowering, not constraining," is a story you're telling, but mathematics constrains and empowers. Limiting it to either side is an ideological move.

But for the general distinction in approaches, which this little problem illustrates, the entire business world disagrees with you, the natural sciences disagree with you, the various branches of engineering disagree with you. — Srap Tasmaner

These are bold claims given how little I've said. You seem crabby and contentious. Are you whipping up bogies to fight against? The OP is about whether mathematical explanations are causal explanations. That shouldn't be a contentious topic.

I think J's OP is interesting. It is something like: if mathematical necessity is not self-supporting, then whence is the necessity derived? — Leontiskos

I think on the whole current philosophy finds the idea of there being necessary truths somewhat uncomfortable. I think we'd rather prefer to be able to stipulate what we think it ought to be, preferably based on evidence. Have a glance at Logical Necessity and Physical Causation.

But, all organic life displays just the kind of functional unity that a painting does, spontaneously. Those patterns most definitely inhere in the organic world. DNA, for instance. — Wayfarer

Can the thought of a pattern in the mind explain the being of a pattern in the world

It is true that we can see many examples of patterns which we find have an aesthetic beauty, and which have arisen spontaneously because of the laws of nature. The web site Natural Form Patterns shows many examples.

We see patterns in the world, but the question is, do the patterns that we see exist in the world or only in our mind. Did patterns exist in the world prior to there being anyone to see them?

A pattern has a unity because of the particular way things are spatially and temporally related to each other, where the whole is more than the sum of its parts.

For example, we can see spatial relationships between the blobs of paint on a Monet canvas creating an aesthetic unity in our minds. However, in the absence of an observer, what is the ontological nature of the spatial relationship between these blobs of paint?

Do spatial and temporal relations ontologically exist in a world absent of any observer?

Because if within a world absent of observers, spatial and temporal relationships had no ontologically existence, then there would be no way of spatially and temporally relating disparate things together, meaning that in the world there would be no patterns, as a pattern can only exist if its parts are somehow related together.

For example, two Monet paintings are alongside each other, "St Lazare Station" and "Water-lilies". We, as observers, can see that there are two different paintings, where the blobs of paint in "St Lazare Station" make one unified whole, and the blobs of paint in "Water-lilies" make a different unified whole.

But in the absence of any observer, what mechanism exists in the world that relates one blob of paint in "St Lazare Station" to another blob of paint in "St Lazare Station" but not to another blob of paint in "Water-lilies"?

In the absence of any observer, how can thing A relate to thing B but not to thing C?

Because if either i) thing A neither related to thing B nor thing C or ii) thing A related to both thing B and thing C, then there would be no patterns existent in a world absent of any observer.

Because if within a world absent of observers, spatial and temporal relationships had no ontologically existence, then there would be no way of spatially and temporally relating disparate things together, meaning that in the world there would be no patterns, as a pattern can only exist if its parts are somehow related together. — RussellA

Well, this is the proverbial can of worms and is far afield from the OP. But my response is that strictly speaking it is completely impossible to imagine a world with no observers. If you do imagine an empty universe, say as it might be before there were sentient beings, that imaginative picture, and even a scientifically-accurate reconstruction of it, contains an implicit perspective or point of view. Because without that perspective, what could you imagine or represent? So, without observers, nothing is related to anything whatever. That is the thrust of the OP The Mind Created World.

do the patterns that we see exist in the world or only in our mind — RussellA

As per above, this question can be asked not only of patterns, but of phenomena generally. One of the books I refer to in the OP above is an important but little-noticed book, Mind and the Cosmic Order, Charles S. Pinter, which is a close examination of these topics. (Little noticed, because the author was a mathematics professor who wrote this book, on cognitive science and philosophy, in his retirement, but because he was not known in that field, it didn't get much attention. But it's an important book in my opinion.)

A pattern has a unity because of the particular way things are spatially and temporally related to each other, where the whole is more than the sum of its parts. — RussellA

There are patterns that appear in inorganic nature, in crystals, snowflakes, larva formations etc. The science behind that is pretty well understood. Organisms embody more than simply patterns as DNA is a code, and codes convey information, which patterns don't. But that is another topic again.

The "effectiveness" of beliefs is tied to the world on the one hand and our own nature on the other (and of course the separation here is not hard and fast).

Suppose bleen is "green and "first observed" during or before 2004," or "blue and 'first observed' after 2004." Could you go walk around where you live and determine what was grue or bleen? Suppose there is a famous green landmark in your town and it got flattened by a tornado in 2006. It was rebuilt with largely with materials salvaged from the original, but has a substantial amount of new material. Is it bleen or grue? What if only small parts of it were replaced each year since 2004?

In terms of what is useful, it seems that inquiry is (usually) going to be best organized according to per se prediction (predication of intrinsic and not accidental properties). See below:

One might push back on Aristotle's categories sure, but science certainly uses categories. The exact categories are less important than the derived insights about the organization of the sciences. And the organization of the sciences follows Artistotle's prescription that delineations should be based on per se predication (intrinsic) as opposed to per accidens down to this day.

This is why we have chemistry as the study of all chemicals, regardless of time, place, etc. and biology as the study of all living things as opposed to, say: "the study of life on the island of Jamaica on Tuesdays," and "the study of chemical reactions inside the bodies of cats or inside quartz crystals, occuring between the hours of 6:00am and 11:00pm," as distinct fields of inquiry. Certain sorts of predication (certain categories) are not useful for dividing the sciences or organizing investigations of phenomena (but note that all are equally empirical).

Of course, there have been challenges to this. The Nazis had "Jewish physics" versus "Aryan physics." The Soviets had "capitalist genetics" and "socialist genetics," for a time. There are occasionally appeals to feminist forms of various sciences. But I think the concept that the ethnicity, race, sex, etc. of the scientist, or the place and time of the investigation, is (generally) accidental to the thing studied and thus not a good way to organize the sciences remains an extremely strong one.

That said, if all categories are entirely arbitrary, the result of infinitely malleable social conventions, without relation to being, then what is the case against organizing a "socialist feminist biology," "astronomy for leap years," and a "biology for winter months," etc ?

They certainly wouldn't be as useful, but that simply leads to the question "why aren't they useful?" I can't think of a simpler answer than that some predicates are accidental and thus poor ways to organize inquiry. We could ground this in the structure of things but it seems it might be simpler to ground it in phenomenology and the quiddity (whatness) of things as experienced. We can think here if how Husserl imagines changes to the noema (object of thought) and considers which changes force it to become an entirely different sort of thing.

Is this meant to be Tarski's view? Surely he didn't talk about what was the case in the world -- only about the correct relations between language and metalanguage. If one language has to be "about the world," then we wouldn't have any logical or mathematical truths at all, or at least that seems to be the necessary consequence. I don't think Tarski intended this. Unless I'm mistaken, he included these kinds of truths in his schema. — J

I think of it as more the Correspondence Theory of Truth, in that a belief is true if there exists an appropriate entity, a fact, to which it corresponds. (SEP - Truth)

I agree that such a Correspondence Theory of Truth draws on ideas developed by Tarski, who was more concerned with mathematical logic than the metaphysics of truth (SEP - Truth).

Tarski's Semantic Theory of Truth (STT) is considered to be a version of Aristotle's Correspondence Theory of Truth, and treats truth as relative, rather than the classical approach of treating truth as absolute.

Considering the schema "snow is white" is true IFF snow is white, Tarski's STT only applies to formal languages, in that "snow is white" is within the object language whilst snow is white is within the metalanguage. Within the modern correspondence theory, "snow is white" is also within the object language whilst snow is white is a fact in the world.

It may well be that we don't know whether an equation is mathematically true or not, but pragmatically, does this matter as long as the equation works. All a scientist wants to know is that an equation works. Even if the scientist did know that the equation was a mathematical truth, this wouldn't affect their use of the equation.

An object in the world emits a wavelength of 700nm and I perceive a red light. I am driving a car, see a red light on a traffic light and know to stop. Have I stopped because I know the truth , that a wavelength of 700nm has been emitted from the traffic light, or have I stopped because I see a red light?

What a cromulent response! :smile:

This just pushes the question back a level -- why is it effective?

Now of course the picture you're painting is a perfectly good one if you're a pragmatist, or believe for whatever reason that metaphysical questions about the correspondence of thought and reality are either incoherent or unanswerable. But I keep pressing you on your use of "accurate representation" -- "accurate" simply doesn't mean the same thing as "effective" or "successful." Wouldn't it make more sense for you (if I've understood your thinking here) to abandon any talk of accuracy or truth?

Interestingly, you've made similar arguments to those of @Philosophim about "effective" truths, and my response to him, just above, is similar to the one I'd make to you. It may well be that scientists don't much care whether equations are true, as long as they work. But philosophers -- and, I'm guessing, a lot of mathematicians -- care very much. We can't take "facts in the world" for granted and go about our business. A theorem (as opposed to an equation that's given a real-world interpretation) isn't described as effective, it's described as true, or at least provable in L. Do you want to abandon that way of talking? If a correspondence theory of truth demands that we do so, I'd argue that it represents a reductio ad absurdum and should be rejected on that ground.

Within the modern correspondence theory, "snow is white" is also within the object language whilst snow is white is a fact in the world. — RussellA

Similarly, I agree that this is a familiar version of a correspondence theory, but it leaves out the option of claiming truth for any facts that are not about the world (unless there's a "world" of math and logic). Is that OK? What would be the point of limiting ourselves in this way?