First:

A Turing machine is not a gadget, but an algebraic structure. In general, a Turing machine cannot perform analogical algorithms. For Instance, to calculate the exact trajectory of a bullet requires some differential equation. Without knowing the existence of differential equations, a human can determine the trajectory of a bullet, simply shooting a gun.

Second:

Computers do nothing without the help of a programmer. Suppose that a human, say T1, builds a program P1 by means of which computers can build their own programs. Of course, it is required the action of T1 together with the program P1. Now, suppose that T2 creates a program P2 by means of which computers can build P1. Of course, they need the action of T2 together with the program P2. Iterating this process computers will be always one step behind humans. In any case, actual computers do not fit in this game, because human brains work with heuristic procedures that not always can be successful. By contrast, to be considered algorithm, a procedure must be always efficient; therefore successful. It is just due to the heuristic nature of human brain, that not every human brain can discover neither quantum mechanics nor Gödel's theorem. It is rather an odd phenomenon. In other words, human success is not predictable, otherwise human beings would be also machines.

Juan-Esteban. I like pretty much your comment, and your special optimism about humans over machines. I almost agree with you. But, dont you think same argument of you can be applied to humans?

I mean, for instance, suppose that one human born in a Borneos island, and he never knew about iPad (and nothing about technology), dont you think he could need the help of other human to know about that? Now suppose an human is responsable to teach technology to all isolated natives living in islands, don't you think this last human need to be taught by other humans?

In the other hand. I accept totally the fact that Turing Machines (as abstract notion of coputation) is not capable to do some things than human can, but don't you think machines can do things than human cannot?

Dear Alberto,

Of course, machines can do things that humans cannot, for instance to lift an object of 1000 kgs.

Please, see the sentence you have written:

"dont you think he (human) could need the help of other human to know about that?"

Can you see the difference? The symmetric sentence to my claim is:

"dont you think he (human) could need the help of a "machine" to know about that?"

Let us consider that if humans need the help of humans, what is the machine role in this game?

The correct argument against my claim would be that machines need the help of other machines, but not by humans.

In any case, perhaps if you read the biography of a human called Ramanujan, you can see how a helpless human has overcame most humans who were helped by many others.

Yeah! The point still the air I think. The really fact is that we know that machines are made by humans, What would happen if we would know who made humans? Dont you think humans were programmed but the human´s creator? then the cicle comes again.

We, as humans, have developed some mental treatments that in general seems like try to reprogram the mind. This fact ebeds the fact that we are programmed. if one human try to reprogram other human, it is clear we, all humans need help of the original programmer, dont you think? Or is the human mind the origin of everything? Selfprogramming? if this last would be true, why we cannot understand the thought at all yet?

Apreciado Alberto,

You have just negated evolution…..

In fact this is the grandeur of human mind: his capability of evolving.

Nevertheless, I hardly can understand your final sentence.

What do you mean by saying "cycle comes again"?

Can you see a cycle in it? In other words: Creator- > human -> machine-> Creator -> human -> machine and so on.

In this cycle, fortunately, I shall become Creator before being a machine, and then I will break the cycle.

Saludos.

Juan-Esteban. I like very much your comment. You are a very smart person with clear ideas.

About you said and ask before, I would ask you first, Do you still believeing on evolution? Do you really think mind has evolved? Don't you think that in primitive times there were and Einstein, or a Van Gogh? It is not a fact about discovering new knowledge about universe and its functionality?

Now, abou the cicle, I meant, you did analogy:

a) P1 needs T1

b) P2, which writes P1, need T2

My cicle, following your analogy:

c)T3 who wants "rewrite" T1 or T2, needs creator

d) then T1 and T2 needs creator to do P1 or P2.

My point is, as you said machines allways need humans is totally correct, just like humans always need creator. Just like machines need their creator.

Then humans are in same situation as machines: they need their programmer.

In this level, if my argument is true, then humans are not superior than machines (in this respect at least).

Dear Alberto,

This message correspond to your later.

Perhaps there are a lot of humans that seem to be programed. This is the aim of marketing industries.

But I can prove that at least some humans can create algorithms without being programed.

Consider that alphabets, like alphabetical order, did not exist 10.000 years ago. Now, most humans can order a set of names alphabetically. This fact proves that at least one human has invented an ordering algorithm with no help.

When a baby starts walking, he does not know how handle his legs. But falling down again and again at last his brain creates a "driver" (algorithm) to this aim. It is a clear example of selection. Every move producing a fall is rejected in future experiences and finally rejecting bad moves, which are performed at random, a proper method is obtained. In this example you can see that the only help can consist of randomness and experience.

Juan-Esteban.

I think you are totally right, randomness and experience are essential factors to exhibit intelligence. But when i say "humans are programmed" I mean that human beings have innerent notions. This last claim of mind is supported (as I can see) four your last comment.

You said "most humans can order a set of names alphabetically" despite the unexistence of an alphabet. You said too: "Every move producing a fall is rejected in future experiences and finally rejecting bad moves".

Who teach to humans about there is an "order" even not having aphabet? Who gives them the concept "bad (moves)"? and in the most basic level, who teach them what is good or bad? all what hurts is bad?

What seems to me, Juan, is like, we (humans) are machine's creators, we are the creation of a creator. Then, just like machines need humans to be, humans need a programmer to be. Then in this analogy, humans, not still being superior than machines, differences are on the reachment and limits. Of course, human brain has wider limits than machines. Then, of course, humans heads are superior about what they can do, because the human's creator is smarter than humans, just like humans are smarter than machines.

Then, we are a kind of machines (once again, superior than computers)

Dear Alberto,

I am a mathematician and in math language what you term cycle is termed chain, sequence, etc. The term cycle denotes circular processes, that is, those sequences that can be represented by means of circles; therefore they have neither beginning nor end.

On the other hand, I do not usualy believe, I prefer to prove, consequently what I know about evolution is what it is proved, by no means, what it is supposed,.

Your creator concept is anthropomorphic. In India 2500 year ago had overcome this concept. But in any case you can believe what you like. Proving hypotheses is another task.

It seems to me that your aim when stating this question is religious rather then scientific. I cannot help you in this view-point.

Juan.

Thanks so much for respond and for your help in my question. It's a big pleasure to know your thoughts about this.

My big problem is that even Turing machines and computability theory, as I can see it is one particular form to see the proving theorems (some purely mathematical fact)- claim (Turing, 1950, 1936) that that is -probably- one explanation how the mind works, they cannot prove anything

Even more, my question -the one that you named religious- still tere for both (maths and computation) areas, where basic principles camed to belong to the mind?

I think is too very hard to think that ramdomness (an essential fact of evolution concept) canproduce this very complex system and concepts (or all the set of axioms used by the mind) and impossible to prove also.

I think, as Gódel proved, Maths suffer the same limitations than computability (and its 4 more known approaches (TM, Recursive funtions, Lamda language and recursivity)), and mind still being superior than maths (for the same arguments you wrote before, like the gun fired and the mind that not need differentian equations to know the trajectory of the bull).

Thanks again.

Dear Alberto

I think that you are mixing mathematics, philosophy and metamathematics, which is partly maths and partly philosophy.

Nevertheless, I would ask you for considering some questions.

1) Suppose that we are the product of a creator and consider the following alternatives.

Are our structures the only possible ones or, conversely, we could have been structured in some different ways?

Of course, each of us can imagine several different structures for our brains and bodies.

If so, why Creator has chosen the actual structure?

There are two possibilities:

1) There is some imperative reason which causes his decision.

2) There is no reason for his choice.

a) If the answer is the first one, the Creator's decisions are caused, hence he is not the first cause.

b) By contrast, if there is no reason or cause, then his decision is obtained at random, therefore our structure is built at random.

The consequence is that, even accepting a creator, if he is the first cause, our structure and existence is obtained at random. The only difference is that, in one case the random phenomena occur in the mind of a Creator, but disregarding the action of a creator, the random process occurs in Universe. Nevertheless, you can believe that Universe is the mind of a Creator, and universe laws his thoughts. As far I know, the latter is what Einstein believed about the creator thouth, but he rejected randomness. Nobody is perfect ....:) :) :)

Juan-Esteban

Have to say that your last logical path is really deep and interesting. Too much.

And made me think that you wrote in a way that you are giving some kind of "intelligence" and "will" to the randomness.

Suppose then that Creator is a fallacy, and randomness was the author of everything and evolution the responsable to made us as we are.

This last fact rise the question what then was the -as you said- "first cause", which I can see is based on a hard supposition of a powerful vision of optimization, inteligence and undirectly the presense of free will.

If this is the fact, don't you think that randomness and intelligence + free will concepts are totally opposite?

Now, suppose that we stilll believing is randomness the mother of everything, Do you think that if i put all the parts of a computer in a container, and I apply electricity and motion to the container, would be possible to get the most simple computer after some millions of years? (supposing that randomeness is possible as rule generator of the amazing parfect complexity of the universe).

Even more, if,a s you said, there are several (maybe infinite) forms of structures, why the our is knowed (and proved) as the best? Do not you think that the answer to this last question would repply how creator (if he/she/it exists) take the decision? I mean, because he/she/it wanted to do as everything it is done. This, for me seems to be pure free will, a characterisc that belong to all alive conscious being, and not to random, cold processes

And, Juan,

Yes, probably seems like im mixturing areas, but Im student of Philosophy of Sciences program in Mexico, with mathematical backgrounds and computation carreer.

As I can say, the last and first step in the scientific travel is the philosophy. That is what all scientist need to prove, think and make theories.

Thank you so much for your comments.

I do not usually say what I cannot prove. If I said something without proof, I say not, simply, I suggest.

I have not any prejudice about intelligence, but I am an intelligent being and I can observe how my mind works. It is absolutely at random. When I am trying to solve a problem, then at random occurs a sequence of ideas in my mind, and I test each of them. If an idea does not pass the adopted suitability criterion, then it is rejected. The process ends when I find one that passes the test.

This is a process quite similar to evolution. At random ideas are occurring and a depredator, namely, the suitability test, kills those that fall the test, until I find the adequate. My studies, knowledge and experience only increment or decrement the occurrence probability of every idea. When for a given problem a solution possess a very great probability it occurs the first. This is the same machinery as evolution.

Do you seriously believe that when you get ill there is not any random process in it? If you seriously believe that your illness is the decision of a creator, then you must accept it; because trying to have some medication would be a rebellion against the creator's decision.

If you can assure to me that you accept every illness, every problem without trying any human solution, then your claim that we are programed by a creator could be taken under consideration.

Of course, you can say that tossing at random the pieces of a watch it is not possible to set it up. But this argumentation is only valid if you assume that all configurations are of the same probability. By contrast, if the probability of the suitable configuration is 0.99 while any other is very small, then the watch is surely obtained. If the universe is as we can see, obviously this configuration and this structure is more likely than others. If you prefer a creator, I prefer 123 creators, because I like the number 123. Everything is a matter of personal preference, but personal preference is not a logic law.

Nevertheless, your are free of believing what you like, but do not confuse "what you like" and "what logics implies".

To interpret properly what I am saying, you must understand that when I write the term random, I mean "those phenomena occurring without any cause determining them". Therefore, in my texts, randomness is synonymous of cause-less. I suppose that you agree this claim, since you have mentioned free-will. If every phenomenon has a cause determining it, there is no room for free-will. By contrast, if there are cause-less phenomena, the consequence is randomness.

Dear Juan-Esteban,

After one day thinking on your arguments, I think you are right in the most part of them. I consider they are logical and reasonables.

So, now, I reglect the creators claim.

Not, from this new clear and clean point, I still findinf my original questions still being in the air.

I still seeing a cotradiction purely logical: Computers always need a human mind to be, because they cannot choose their own programmation or language. But humans cannot either. Then both still suffering same limitation, then, they are in this sense, equivalent and there is not superiority of one over the other.

Thanks so much for your comments.

Dear Levente Kriston,

Thank you for the very interesting information you have provided.

Now, let me to write some ideas about Gödel.

What Gödel believes or claims does not matter. What I take under consideration is what Gödel proves.

Every logical system is built upon some axioms. Depending on those axioms accepted by Gödel, he can obtain some results or the opposite ones. This is the structure of axiomatic method.

But the real world need not accept the same axioms as Gödel. To know what are those axioms ruling real world it must be investigate the real world itself. Thus, the logic of real world must be an empiric science.

Computers are the result of the action of some intelligent systems (human beings); by no means are their cause. We must not confuse cause and effect. Even a human baby can dream. Dreams are the seed of creativity. While computers are not able to dream, no creativity can be in them.

William Jackson.

I agree all with you, Gödel's theorem applies just for axiomatic systems.

If we consider that is a fundamental true, what would you think is the best way to study the mind? (Considering formal system's limitations)

William and Levente Kriston, It seems a great suggestion what you said.

But you mean using artificial Neural Networks or brain direct study? I ask this because when I was talking about this with my assesor, the said something I think is very smart and interesting:" Even Artificial Nural Networks can ve reduced to Turing Machines" (since they are practically always implemented in a computer) and as one Turing Machines system, they cannot be in the same level than the mind.

Thank you so much for your comments

Each formal system has its limits. The location of this limit varies from formal system to formal system. Mathematics is not a single formal system, it is a manifold of such systems. It is probably not too optimistic to assume that within mathematics fruitful systems for analyzing thoughts can be formulated.

In natural sciences (to which studies on thoughts belong) the limits of formal systems never come from the problems which Goedel studied (and solved): these arise from the infinity of objects under consideration, a situation that never happens in nature. In natural sciences the limitations shine up in failure to agree with experimental facts. The normal reaction of scientists then is to change (radically or evolutionary) the formal system. Physics, chemistry, biology, ... all are full of examples of this.

In conclusion: Formal systems are creations of the human mind. If designed with some preliminary insight into the working of the 'real world', formal systems have the potential to sharpen and direct this insight (into the working of the 'real world' ) and also to suggest improvements of the formal systems we started from. Even the concept of formality will change (see history of mathematical logic) in this process.

Ulrich Mutze.

I agree totally with you.

Can you suggest something how natural sciences can enter in thoughts realm?

To the majority of natural scientists thoughts are processes going on in brains. Thus all research on brain activities are relvant here. From a more theoretical point of view also research on 'neural networks' (in the sense of mathematical models) should be considered.

Ulrich Mutze,

Yes, actually those are common approaches to study the brain system, but I have a hard conflict with those. Maybe you would have a guide comment about.

Firstly, I think direct brain experimentation (reading of signals, paths of signals using specific stilumuli) say much in a physical level of the brain, but not much about the thought itself: logic, what is proccessed and how, information representation and more issues.

Secondly "articicial Neural Networks" are computer implementations, then suffer of gödel limitations (by his theorem).

What do you think about? Any suggestion?

@Alberto, observing the physical brain activity of a person who is solving of a specific task (e.g. computing 93*3 in one's mind) may give hints concerning how the thought and its physical realization are related.

Don't care about 'Gödel limitations'. Some people have blown up the nice mouse to a specter and too many people follow them without having looked deep enough into the matter. Somewhere on RG I analysed the 'Gödel limitations' in the context of the famous Goldbach conjecture; this could help you to put things into the right perspective.

Classical logic is consistent but "explosive" ... one contradiction implies EVERYTHING. There are other (necessarily slightly weaker) systems of logic which are paraconsistent, that is, can tolerate some contradictions without blowing up. (One of my favorites is First Degree Entailment but there are many others.)

Why do I bring this up here? Because Gödel's proof works by constructing a contradiction, and in some systems of paraconsistent logic, the proof fails. This suggests that a complete theory of arithmetic might be possible, but ONLY if it allows some contradictions (i.e. is inconsistent) and is based on a paraconsistent logic rather than classical logic.

Howard,

I found your words very logical, I didn´t know about that kind of logics. I will look for it, in fact, i t hink that would solve my problem with formal maths.

I think this problem brought by my backgrouds as matematician.

Thank you so much.

Alberto, An Introduction to Non-Classical Logic by Graham Priest is a good book to start with if you want to get a sense of what the various non-classical options are. Or just google "paraconsistent".

Howard,

I got the book, Thank you a lot!

Have confess that as my backgrouds are in maths I always was customed on to think about formal system and the power of them to explain the universe. 1 year ago i knew the work of several researcher who question the mathematical work in the sense that there must be a way to tolerate contradictions because the world is not formal at all. This book seems very interesting.

Thanks again.

Best to all involved in this conversation. And thanks too

It is inaccurate to say, "Gödel´s theorem proved that formal systems have limits on what they can express, but it is clear than the mind can calculate what formal systems cannot."

Godel's theorem did not make claims about what formal systems can express. It asserted that no formal system in logic can be both consistent and complete. Consistent systems cannot support valid proofs taht lead to contradictions. Complete systems are those within which any true assertion can be proven.

Godel's proof was based on an example assertion in logic, similar to "This assertion is unprovable," that was both obviously true (outside the given formal system) and clearly unprovable within the system.

To my knowledge, Godel did not try to relate his theorem to the capabilities of the mind. If the mind were considered a formal system, then his theorem would apply.

The kind of thing that the mind can calculate that "formal systems cannot" does not include validating proofs of assertions in logic. The mind can do lots of things that formal systems can't, like calculate esthetic preferences from sensory inputs and bodily impulses.