Philosophy Reading Group / Kant & Math

[apeiron]

(maybe you consider this compromising with the mainstream) building bridges between dynamicists and computationalists.

Yes. Simply put, I was finding that many wanted to "find the truth" of a dynamic view by reframing it in computational terms. But then I came away from debates with Pattee, Salthe and others wanting to do the opposite - to find a dynamic way to frame the truths of computationalism. Define an organic logic to complement the more familiar mechanical logic, so to speak.

Other grad students of Pattee, like Peter Cariani, also had more radical ambitions and consequently, I would say, struggled to make it in academia.

 

[nikman]

Frankly, I see no important difference between Rocha and Cariani ...

 

[nikman]

Just re-read Cariani's "Symbols and Dynamics in the Brain" (linked from Rocha's website) and Rocha's papers linked above ...

Are you perhaps factoring in some personal dimension here?

 

[SixNein]

We might, indeed at first suppose that the proposition 7 + 5 = 12 is a merely analytical proposition, following (according to the principle of contradiction) from the conception of a sum of seven and five. But if we regard it more narrowly, we find that our conception of the sum of seven and five contains nothing more than the uniting of both sums into one, whereby it cannot at all be cogitated what this single number is which embraces both. The conception of twelve is by no means obtained by merely cogitating the union of seven and five; and we may analyse our conception of such a possible sum as long as we will, still we shall never discover in it the notion of twelve. We must go beyond these conceptions, and have recourse to an intuition which corresponds to one of the two− our five fingers, for example...​

Do you agree with Kant here? Do you agree that our conception of the number 12 depends to some degree on experience and isn't something we can derive rationally?
/QUOTE]

I disagree with Kant.

All you need is rules.

 

[JoeDawg]

All you need is rules.

But where does one get the rules?

Without some foundation, the selection process is random.

 

[SixNein]

But where does one get the rules?

Without some foundation, the selection process is random.

The foundation is the rules. We could invent our own rules and create our own numbers.

The letter @ is a number.
Every number has a subsequent number.
No numbers share the same former number.
@ does not come after any number.
Any property that belongs to the number @, and any property that belongs to subsequent numbers, belongs to all numbers.

Using the above rules, I have created the following list:
@, A, B, C, D, E, F, G, ... N

But you need more rules to do anything with it.

In a basic nutshell, the rules of the number serve as the foundation.

 

[Sorry!]

The foundation is the rules. We could invent our own rules and create our own numbers.

The letter @ is a number.
Every number has a subsequent number.
No numbers share the same former number.
@ does not come after any number.
Any property that belongs to the number @, and any property that belongs to subsequent numbers, belongs to all numbers.

Using the above rules, I have created the following list:
@, A, B, C, D, E, F, G, ... N

But you need more rules to do anything with it.

In a basic nutshell, the rules of the number serve as the foundation.

The question was WHERE do you get the rules, not how can we formulate them. So explain why you have chosen those rules above... and what parameters we should use to select further rules?

 

[SixNein]

The question was WHERE do you get the rules, not how can we formulate them. So explain why you have chosen those rules above... and what parameters we should use to select further rules?

You obtain the rules from thinking and asking questions. You could create different rules that are weaker or stronger, and the nature of your numbers would change depending upon the rules. In this example, I just spit out a rehash of Peano's axioms.

 

[SixNein]

Most?

Most is correct.

The continuum hypothesis is independent of our current system, so we cannot solve it without creating new axioms. Quite a few different problems are independent.

 

[JoeDawg]

You obtain the rules from thinking and asking questions. You could create different rules that are weaker or stronger, and the nature of your numbers would change depending upon the rules. In this example, I just spit out a rehash of Peano's axioms.

How do you measure strength of a rule?

 

[SixNein]

How do you measure strength of a rule?

Through the means of consistency.

I personally think Kant is off base. Counting is one human field that can be found in nature. A wolf can count, and it doesn't use its fingers. The concept of numbers is extremely primitive even though the definition of numbers is quite complex. Although our counting abilities are developed through external means, life could find the means to count through thought.

 

[JoeDawg]

Through the means of consistency.

Consistency demands that you have at least 2 axioms, how do you select the first one?
This is where experience comes in. Even the very idea of 'consistency' comes from experience. We value consistency because of its value in predicting outcomes which are beneficial to us. We evolved this ability, because its benefitial to survival.

Although our counting abilities are developed through external means, life could find the means to count through thought.

And a millions monkeys typing on keyboards for an infinite amount of time could write the complete works of shakespeare. But 99.9999...% of what they churn out would be nonesense, and some would be 'consistent' but not resemble reality.

Math was invented by generalizing experience.

 

[SixNein]

Consistency demands that you have at least 2 axioms, how do you select the first one?
This is where experience comes in. Even the very idea of 'consistency' comes from experience. We value consistency because of its value in predicting outcomes which are beneficial to us. We evolved this ability, because its benefitial to survival.



And a millions monkeys typing on keyboards for an infinite amount of time could write the complete works of shakespeare. But 99.9999...% of what they churn out would be nonesense, and some would be 'consistent' but not resemble reality.

Math was invented by generalizing experience.

Thought is an experience. Mathematics is a manor of symbolic thinking.


When was math invented exactly? Mathematical abilities can be found in other species.

 

[JoeDawg]

Thought is an experience. Mathematics is a manor of symbolic thinking.

Thought is qualitatively different from sense experience.

When was math invented exactly? Mathematical abilities can be found in other species.

Hard to say, but geometry goes back at least as far as the Ancient Egyptians. Before writing, I'm not sure one could even have a formal system.

Mathematical ability derives from logical ability, which comes from observing the world.
If you've ever watched a baby learn, you'll see how they start to form logical patterns.

Formalized mathematics is different from ability however. Simple counting doesn't really require a formal system. Few, some, many... is counting.

A formal system of math probably requires writing.

 

[SixNein]

Thought is qualitatively different from sense experience.

Hard to say, but geometry goes back at least as far as the Ancient Egyptians. Before writing, I'm not sure one could even have a formal system.

Mathematical ability derives from logical ability, which comes from observing the world.
If you've ever watched a baby learn, you'll see how they start to form logical patterns.

Formalized mathematics is different from ability however. Simple counting doesn't really require a formal system. Few, some, many... is counting.

A formal system of math probably requires writing.

In the particular situation we are discussing, the person probably can't do much math. In fact, a person isolated from the rest of humanity could not do much either. But I believe the person could grasp the concept of numbers without senses. Could you tell if you had more than one thought?

 

[JoeDawg]

Could you tell if you had more than one thought?

I think this is one of the problems faced by those who want to create an AI. I think our consciousness is a function of interacting experiences from multiple stimuli. Creating a brain in box, or brain in a vat, means there is really only one source of stimuli. If we want to mimic intelligence, I think we need to include a variety of input sources. A robot with eyes, ears... a sense of touch...etc..

But to answer your question, I think you would have to have a need to distinguish between thoughts, and that need would undoubtedly rely on an external influence.

 

[wofsy]

A pretty big problem from what I have read. I think you're idealizing mathematics, like the ancient greeks did with geometry. It all happens in the human brain. And different cultures have developed different mathematical systems. Ours has simply absorbed all the aspects we find most useful.

Math is in large part analytic, but its axioms and basic logic are derived from experience. All logic comes from how we see the world working. Math is just a way to represent and predict experience using highly abstract language.

Axioms are little more than assumptions, or constraints. And those constraints are based on our experiences in the world.

A couple unrelated thoughts: They are meant as talking points not as assertions of anything that I believe.

- in some sense everything happens in the human brain. Mathematics and experience are inescapably unified.

- Logic is intrinsic. If it were not we would not be able to think.

- "Derived from expeience" is a vague idea. Derived has no rigorous definition.

- Math is not "just" a way to represent and predict experience". Here is a quote from Felix Klein's introduction to his book, Riemann's Theory of Algebraic Functions and their Integrals."

"He(Riemann ) had in mind far more general methods of determination than those we employ in the following pages;methods of determination in which physical analogy ...fails us."

- Experience suggests an underlying mathematical reality but that reality supercedes direct experience

 

[apeiron]

A couple unrelated thoughts: They are meant as talking points not as assertions of anything that I believe.

- in some sense everything happens in the human brain. Mathematics and experience are inescapably unified.

- Logic is intrinsic. If it were not we would not be able to think.

- "Derived from expeience" is a vague idea. Derived has no rigorous definition.

- Math is not "just" a way to represent and predict experience". Here is a quote from Felix Klein's introduction to his book, Riemann's Theory of Algebraic Functions and their Integrals."

"He(Riemann ) had in mind far more general methods of determination than those we employ in the following pages;methods of determination in which physical analogy ...fails us."

- Experience suggests an underlying mathematical reality but that reality supercedes direct experience

If you want to demystify and make rigorous the relationship between minds and worlds, Rosen's modelling relations work serves as a good template...

http://www.panmere.com/?page_id=18

http://www.osti.gov/bridge/servlets/purl/10460-5uGkyu/webviewable/10460.pdf