What LEM is not

[godelian]

ChatGPT said:

3. Potential Justifications for Physicalism

Mathematics as a Cognitive Construct: Another physicalist stance might propose that mathematical ideas are constructs of human cognition, shaped by our interactions with the physical world. This viewpoint could explain why mathematics seems so effective in describing natural phenomena without requiring that mathematical entities exist outside of physical reality.

Some mathematics and logic are clearly built into our biological firmware for immediate reasons of survival. This is also the case for animals:

ChatGPT said:

Do animals use mathematics?

Yes, some animals do exhibit behaviors that suggest they use basic mathematical concepts. For example:

- Counting: Certain species, like some birds and primates, can count or have a sense of quantity. Studies have shown that they can differentiate between different amounts of food or other objects.

- Spatial awareness: Animals like bees and ants demonstrate an understanding of geometry and spatial relationships, which can be seen in their navigation and foraging behaviors.

- Patterns and sequences: Some animals can recognize patterns and sequences, which can help them in hunting or foraging strategies.

- Social structures: Animals like elephants and dolphins show complex social interactions that may involve an understanding of social hierarchies and relationships, which can be seen as a form of mathematical reasoning.

While these behaviors may not involve mathematics in the way humans understand it, they do suggest a capacity for quantitative reasoning and problem-solving.

Do animals use logic?

Yes, many animals exhibit behaviors that suggest they can use some form of logic. For instance, studies have shown that certain species, like primates, dolphins, and corvids (like crows and ravens), can solve problems, make tools, and understand cause-and-effect relationships.

For example, crows can figure out how to use sticks to extract insects from tree bark, indicating an understanding of the relationship between the tool and the task. Additionally, some animals can anticipate outcomes based on past experiences, suggesting a form of reasoning.

While their logical processes might not be as complex as human logic, there's definitely evidence that various animals can think critically and adaptively in their environments.

Informal mathematics and informal logic as built into our biological firmware were historically the predominant alternative:

https://en.wikipedia.org/wiki/Informal_mathematics

Informal mathematics, also called naïve mathematics, has historically been the predominant form of mathematics at most times and in most cultures, and is the subject of modern ethno-cultural studies of mathematics.

Informal mathematics means any informal mathematical practices, as used in everyday life, or by aboriginal or ancient peoples, without historical or geographical limitation. Modern mathematics, exceptionally from that point of view, emphasizes formal and strict proofs of all statements from given axioms. This can usefully be called therefore formal mathematics. Informal practices are usually understood intuitively and justified with examples—there are no axioms.

There has long been a standard account of the development of geometry in ancient Egypt, followed by Greek mathematics and the emergence of deductive logic.

The systematic formalization of mathematics and logic started in Greek antiquity. That is also the point in history at which the awareness arose that in physical reality we can only see some small vague part of a much larger abstract, Platonic world:

https://en.wikipedia.org/wiki/Allegory_of_the_cave

In the allegory, Plato describes people who have spent their entire lives chained by their necks and ankles in front of an inner wall with a view of the empty outer wall of the cave. They observe the shadows projected onto the outer wall by objects carried behind the inner wall by people who are invisible to the chained “prisoners” and who walk along the inner wall with a fire behind them, creating the shadows on the inner wall in front of the prisoners. The "sign bearers" pronounce the names of the objects, the sounds of which are reflected near the shadows and are understood by the prisoners as if they were coming from the shadows themselves.

Only the shadows and sounds are the prisoners' reality, which are not accurate representations of the real world. The shadows represent distorted and blurred copies of reality we can perceive through our senses, while the objects under the Sun represent the true forms of objects that we can only perceive through reason.

We cannot see the complete abstract, Platonic world just by observing physical reality. We can only perceive it through blind and pure reason. Physicalism is therefore an attempt to throw us back before the discovery of Plato's cave. If you do not believe that the abstract, Platonic world exists, then you will obviously not be able to see it.

https://en.wikipedia.org/wiki/Philosophy_of_mathematics

Kurt Gödel's Platonism[34] postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly.

I personally do not believe that the ability to see the abstract, Platonic world of mathematics is some special kind of talent or intuition. You just need to be open-minded enough. However, people who have been thoroughly indoctrinated in physicalism will not see it. It is not that they do not have the talent to see it. They simply do not want to see it.

[Magnus Anderson]

Physicalism fails because it cannot deal with the abstract, platonic phenomena that do not have a counterpart in the physical universe but which are essential to explain the ones that do.

For example, the 18th century is replete with philosophers and mathematicians who rejected calculus because infinitesimals were not physically "real" as they do not map to anything in physical reality:

[..]

I am afraid you're distracting yourself by confusing my positions with those of others.

You're trying to shoehorn me into one of the common philosophies of mathematics: physicalism, platonism and formalism.

That's not a good idea.

I am not rejecting infinitesimals. I see absolutely no reason to do so. It's not something that follows from the way that I think.

My claim was merely that concepts, such as the concept of infinitesimal, are physically real.

There is a difference between concepts and quantities. A quantity is not a concept. The concept of a quantity is a concept. Concepts are meanings attached to symbols and they represent the rules that determine what can be represented by these symbols. The concept attached to the symbol "4" tells us that we can only use that symbol to represent the quantity that is 4. That's not a quantity. It's a set of rules.

As such, the statements "Concepts exist" and "Quantities exist" are two different statements meaning two different things.

Then you have to consider that there are more than one realm of existence. There are the physical realm of existence, the conceptual or platonic realm of existence and the oxymoronic realm of existence.

The physical realm of existence refers to what actually physically exists.

The conceptual realm, or the platonic realm, refers to what can be conceived or imagined regardless of whether it physically exists. Whatever exists in the conceptual realm, does not necessarily exist in the physical realm, but it can exist. Whatever does not exist in the conceptual realm has no possibility of existing in the physical realm. The latter are things such as square-circles. They only exist in the oxymoronic realm.

In mathematics, when we say that a quantity exists, we're saying that it exists in the conceptual or platonic realm. We're not saying that it physically exists.

A quantity is said to physically exist if it can be found in the actual physical existence. For example, if there are three apples on my table, then the quantity "three" physically exists. That's what it means. Quantities aren't physical objects, the way rocks are, but they are nonetheless aspects of physical existence.

The concept of quantity that does not physically exist is potentially useless -- why talk about it? -- but that's not necessarily the case. Circles may not exist anywhere in the physical world but they are still useful as an approximation for real life shapes.

And when it comes to infinitesimals, one has to understand that they are fractions, like "1/2". Fractions aren't proper quantities. They are actually relations between quantities. A fraction answers the question, "How many times a number is larger than some other number?" But rather than telling us that the number is larger, it tells us that it's actually smaller. "1/2" means "two times smaller". "1/3" means "three times smaller". And so on. The same goes for infinitesimals. Thus, if you want to determine whether or not infinitesimals physically or conceptually exist, you have to focus on the relation between quantities.

As an example, the set A = { 1 } is infinitely many times smaller than the set N = { 1, 2, 3, ... }. More specifically, if we use the symbol "infN" to denote the number of elements in the set N, then the answer to the question, "How many times is A larger than B?", the answer is "1 / infN" times. "1 / infN" being an infinitesimal. The would be a proof that infinitesimals exist in the conceptual realm. And that might be enough for mathematics.

If you want to prove that infinitesimals exist in the physical realm, it should be easy if you happen to already believe the space is infinite in all directions. If you're sure that the length of the universe in one direction is "infN" meters, then it follows that the number of times one meter is larger than the length of the universe in that direction is "1 / infN" times. That's an infinitesimal and an actual one.

But the question is, do you really have to do that? Just to appease the physicalists?

[Skepdick]

There is a difference between concepts and quantities. A quantity is not a concept. The concept of a quantity is a concept.

Conceptually - there is zero difference. Least you want to argue that the concept of a concept is a concept; but the concept itself is not a concept.

You recognize a quantity because it's an instantiation of the concept.

You are arguing for a Platonic view.

Our understanding of quantities is inseparable from our conceptualization of quantities.

[Magnus Anderson]

Conceptually - there is zero difference.

That's true in Skeppie McDickie's fantasy world.

Least you want to argue that the concept of a concept is a concept; but the concept itself is not a concept.

Keep such arguments for yourself. Both the concept of a concept and the concept itself are concepts. But quantities are not concepts.

A concept is merely a set of rules attached to a symbol that determine its meaning, i.e. the set of all conceivable phenomena that can be represented by the symbol.

There is a human being, a physical object that occupies a portion of space. And then there is the concept of a human being, a set of rules that determine the meaning of the term "human being". Two very different things.

The same applies to quantities. The number of people who currently live on planet Earth is an example of a quantity. That's very clearly not a concept since it's not a set of rules determining the meaning of a symbol.

You recognize a quantity because it's an instantiation of the concept.

Human beings are instances of the class named "human being". In the same exact way, quantities are instances of the class named "quantity". The two classes are concepts: the concept of a human being and the concept of a quantity. But neither human beings nor quantities are classes themselves.

[Magnus Anderson]

But it has to be there..

And that's your own mistake.

One of your core problems is that you're a literalist, someone who takes everything at face value. The consequence of that is that you constantly misinterpret others.

Since I am the one who wrote the expression "2-5=-3" in the first place - I know exactly what was; and especially what wasn't between the lines.

If that statement of yours was expressed in your own version of standard mathematical language, you should have notified us.

Preferrably, you should have translated that statement in the standard mathematical language so that we can all understand you.

In fact, before you asked your question, I was talking about mathematical expressions written in THE STANDARD MATHEMATICAL LANGUAGE.

I was talking about what "2 + 2 = 4" means in the standard mathematical language.

Then you came in and asked, "And what does 2 - 5 = -3 means?"

You're either a liar ( with the intention to disrupt and destroy thought ) or a mentally challenged individual.

The problem is this basket of yours with "x-2" apples. Where x=1.

An impossible operation. That's why it says, "For every sufficiently large x." There are other, much better ways, to word that statement. You should be able to see it yourself if you're intelligent enough.

[Skepdick]

Keep such arguments for yourself. Both the concept of a concept and the concept itself are concepts. But quantities are not concepts.

Uhuh. So you have no conception of a quantity? No wonder you don't understand anything...

A concept is merely a set of rules attached to a symbol that determine its meaning, i.e. the set of all conceivable phenomena that can be represented by the symbol.

So you have no conception of rules? Could you quantify the number of rules attached to the concept of a "rule"?

Human beings are instances of the class named "human being". In the same exact way, quantities are instances of the class named "quantity". The two classes are concepts: the concept of a human being and the concept of a quantity. But neither human beings nor quantities are classes themselves.

"Human being" is a concept. You happen to identify with it, but there's no rule which says you can't define yourself as "human being"; or abstract yourself as a "human being".

[Magnus Anderson]

So you have no conception of a quantity?

Explains to us what you mean when you say thas someone has no cnoception of a quantity.

Could you quantify the number of rules attached to the concept of a "rule"?

Could you tell us your GPS coordinates?

[Skepdick]

Explains to us what you mean when you say thas someone has no cnoception of a quantity.

...quantities are not concepts.

Could you tell us your GPS coordinates?

Yes. I can. 33.9221° S, 18.4231° E + radius 10km

Now... Could you quantify the number of rules attached to the concept of a "rule"?

[godelian]

I am afraid you're distracting yourself by confusing my positions with those of others.

You're trying to shoehorn me into one of the common philosophies of mathematics: physicalism, platonism and formalism.

Without further clarification, the following does sound physicalist:

A proposition is an idea that a portion of reality exists in certain state.

It depends on what you mean by "reality". The default interpretation for the term "reality" is "physical reality".

[Magnus Anderson]

Yes. I can. 33.9221° S, 18.4231° E + radius 10km

How can I know that's true?

Now... Could you quantify the number of rules attached to the concept of a "rule"?

Explains to us what you mean when you say thas someone has no cnoception of a quantity.

[Skepdick]

Yes. I can. 33.9221° S, 18.4231° E + radius 10km

How can I know that's true?

You know exactly how. Exhaust the search-space.

If you don't find me - it's false.
If you find me - it's true.

Explains to us what you mean when you say thas someone has no cnoception of a quantity.

Somebody who says "quantity is not a concept". e.g you.

[Magnus Anderson]

Without further clarification, the following does sound physicalist:

A proposition is an idea that a portion of reality exists in certain state.

It depends on what you mean by "reality". The default interpretation for the term "reality" is "physical reality".

When I say "reality", I mean "the state of the universe at every point in time -- past, present and future -- together with the laws that govern it".

When I say "portion of reality", I mean "an isolated aspect of reality; a small part or a segment of it".

Every proposition has the form "A is B" where A is a portion of reality and B is a description of that portion of reality.

A portion of reality can be any portion of physical space at any point in time ( past, present or future. )

It can also be any physical object at any point in time ( past, present or future. )

It can also be a relation between the state of a portion of space at one point in time and the state of that same portion of space at another point in time.

A portion of reality can also be the volume of a physical object, its shape, its color, its location and so on.

A portion of reality can also be the number of physical objects of certain kind within certain region.

It can also be a relation between the volume of one physical object and another, e.g. how many times one physical object can be contained within the other.

A portion of reality can also be a law that governs certain phenomena.

And so on.

[Magnus Anderson]

You know exactly how. Exhaust the search-space.

And before I do that, I have to conclude that you don't exist?

If you don't exist, why should I bother conversing with you?

Explains to us what you mean when you say thas someone has no cnoception of a quantity.

Somebody who says "quantity is not a concept". e.g you.

Try again.

[Magnus Anderson]

Exhaust the search-space.

Have you ever opened your skull to if there's a brain somewhere inside it?

If not, does that not mean you have no brain?

[godelian]

Without further clarification, the following does sound physicalist:

A proposition is an idea that a portion of reality exists in certain state.

It depends on what you mean by "reality". The default interpretation for the term "reality" is "physical reality".

When I say "reality", I mean "the state of the universe at every point in time -- past, present and future -- together with the laws that govern it".

When I say "portion of reality", I mean "an isolated aspect of reality; a small part or a segment of it".

Every proposition has the form "A is B" where A is a portion of reality and B is a description of that portion of reality.

A portion of reality can be any portion of physical space at any point in time ( past, present or future. )

It can also be any physical object at any point in time ( past, present or future. )

It can also be a relation between the state of a portion of space at one point in time and the state of that same portion of space at another point in time.

A portion of reality can also be the volume of a physical object, its shape, its color, its location and so on.

A portion of reality can also be the number of physical objects of certain kind within certain region.

It can also be a relation between the volume of one physical object and another, e.g. how many times one physical object can be contained within the other.

A portion of reality can also be a law that governs certain phenomena.

And so on.

So, your definition for the term "proposition" is indeed physicalist.