Yes. That will work fine if the criterion for their order can't change. But you have posited that they can change how much they eat. — Ludwig V

Hopefully I didn't. All the dogs eat exactly a defined amount of food different from any other dog, not less, not more.

(BTW, if someone is puzzled why I did choose Plato and not Aristotle, it's because we learn about Zeno of Elea through Plato's writings. Of course the objection is more Aristotle, but the time gap is even bigger between Zeno and Aristotle. Some reality to a story with Greek goddesses meeting philosophers. :joke: )

Zeno is right. Not by reason of counting. Rather, by rule #2, the one that eats "the most" and the one that eats "the least" are conceptual quantities that differ from any other quantities already given.

It is always valid to say "there is at least one dog that eats the most" and "there is at least one dog that eats the least". — L'éléphant

I agree. I had identical thoughts, but I couldn't find the perfect words to express them as you did. :sweat:
Yes, I am one of the 60% of voters that chose the second choice. — javi2541997

Absolutely fantastic! :grin:

And yes, #2 gives us the opportunity to do this. Not by reason of counting as obviously both of Zeno's dogs are literally uncountable. Notice how different the story would come if Athena would have said: "By using this dog and what it eats, put all the other dogs into order."

As with the story, I argue that this has been a real problem that Math at least over 2400 years.

Now the thing is that our confusion hasn't stopped us for 2400 years as we are already using both of Zeno's dogs in a variety of way in mathematics, but since this question isn't answered how Plato's and Zeno's dog should logically coexist, we simply have different names. In the case of the dog that eats the least, usually called an infinitesimal, we have non-standard number systems (surreal and hyperreal numbers) and Non-standard analysis.

And not only that. When you think that in the story "the least eating dog's meal" < "every other dogs meal", think about how Dedekind cuts are used to define real numbers. The name's cut comes from using greater than and less than signs ">" and "<".

Yet the other Zeno's dog and it's problematic diet comes immediately into question when people tried to form the foundations of mathematics like with set theory. The great Cantor understood what in the story is Plato's argument. For example, when he had the idea of sets having Power set, which are bigger than the set itself, he understood how the power set for "set of all sets" is a bit problematic. And thus Cantor's set theory is hierarchial. Yet what is interesting (and what I can thank the PF forum leading me to a great book) is that Cantor didn't reject the notion of Absolute Infinity, or what in the story would be the dog that eats the most. As he couldn't describe this infinity, he thought it was something that God would know. Or God. Something that for a deeply religious person is something important (unlike for the many reading Cantor's texts). Still, with Frege's simple idea of Basic Law V invites immediately the other of Zeno's dogs to the picture.

As you can notice, in the poll of the story I didn't actually ask if Plato is right or both are right. And this is the real question for mathematics: how can both Zeno's dogs and Plato's dogs coexist in peace?

I'm sorry. I just don't follow this. Is there a typo somewhere? — Ludwig V

Nope, this is basically Plato's argument in the story: increasing the food or decreasing the food size you always get a new dog's meal. So he reasons that there cannot be the dog that eats the most or the least. Well, in finite dogs this holds true, but notice that Zeno's dogs aren't finite. Hence if you add to Plato's dog the amount the dog that eats the least, you would still have Plato's dog eats. Addition and substraction breaks down, or simply is confusing. The best example of this is the Hilbert Hotel, when it comes to the dog that eats the most.

"Plato's dog" is the dog that Plato chose. Let's call the dog that eats less than any other dog "Dog One" and the dog that eats more than any other dog "Dog Two". and the dog that gets less than Dog One "Dog Three". — Ludwig V

But then "Dog One" would eat more than "Dog Three", so how could it be the one that eats the least? Remember, it eats less than any other Dog. I think here it's easier to say that the dog Plato picked up is "Dog One", if you think about it.

That won't work if you start messing about with how much they eat. — Ludwig V

In Mathematics there is this well ordering theorem, so we can assume we can put them into order. Plato did it with his Dog 1, then on one side the dogs that eat more, and on the other side the dogs that eats less.

Isn't your problem with dogmatism, or a misuse and/or misunderstanding of science/positivism, instead of with science/positivism itself? — Philosophim

Yes, absolutely! All the various philosophical schools of thought have contributed each their way. Even if I criticize reductionism and favour the idea of more-is-different, there's a place for reductionism. Yet positivist are the one's that can quite easily fall into that dogmatism.

Perhaps only scientism can be defined so negatively, that is basically something derogatory. (Note that Tarskian referred to scientism, not science.)

Oh, you're imagining that you have discovered a previously unknown manuscript. Who wrote it - Plato, Zeno, Themis, Athene, Zeus? Or a rat, skulking in a corner. — Ludwig V

Well, the reasoning of the Eleatic school isn't this, but do notice that Zeno's paradoxes are handled by limits ...or infinitesimals. So it begs the question.

You cannot take Plato's dog, add the food of the dog which eats less than every other dog, and then get more than Plato's dog eats. If you would get a different amount of food, then that could be divided even smaller portions and the dog that eats the least wouldn't be the one eating the least. So the question here is: while the dog that eats less than any other dog, does this the definite it separate from all other dogs?

My problem is with positivism and scientism. I find these ideological beliefs to be very dangerous.
— Tarskian

I find this point more interesting. Why? — Philosophim

I would agree with @Tarskian, especially a mix of both can be harmful, because one can come to be so dogmatic that one starts to think that model or theory of reality is far more real than just the reality itself. And this dogmatism leads people forgetting that scientific theories are only models of reality. You don't care how real life is different from the scientific model, the model itself is right.

Scientism itself can be viewed as a derogatory remark, but positivism itself isn't so bad, if you don't use it too much and aren't open to other thoughts. For example let's think about Comte's law of three stages, where first people believe in myths and magic, then the society transform's to a transitional metaphysical state and then finally, it becomes positivist society based on scientific knowledge.

That is an interesting idea, but is it a law? Will this really evidentially happen because it's a law? It's a building block of positivism itself, sure, but is it a building block of reality?

You may like to consider the possibility that Zeno's dogs don't exist. (After all, he told lies about Achilles and the tortoise.) — Ludwig V

Did he? Or did he try to make an counterargument to Plato? During the time, you tried to make questions that the one answering you would make the argument. So could it be that Zeno was arguing that by Plato's reasoning you get into the silly ideas like the Achilles cannot overtake the tortoise. Or the Arrow cannot move. Remember, the story is told by Plato, not by a third actor.

If there's isn't that dog that eats the least, then Achilles passes the tortoise. But if you think about Plato's dog and that it can define everything else, then you get the problem. And before you think Zeno's dog that eats the least still isn't a dog, think about it another way:

Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are the reciprocals of one another.

In Abraham Robinsons nonstandard analysis that dog that eats the least exists and is fine.

So why don't we have it in our standard real number system?

Because of Plato. (Or at least, because of Plato in this story)

I would suggest the transfinite system as a home for the other dog - since it's the last one and w (omega) is the limit of the series — Ludwig V

After all, each dog can be counted and the counting can continue for as long as there are any dogs that have not been counted. — Ludwig V

Your second statement goes with the lines of Plato then. Poor of Zeno's dogs.

And with the transfinite, Cantors set of theory of ever larger and larger infinities, it could be argued that this is somewhat similar method to adding. But as I learned from the forum (and got hold of a great book about Cantor) he did think also about Absolute Infinity, but the deeply religious mathematician held it for only God to know.

That may very well be in violation of Carnap's diagonal lemma:

"For each property of logic sentences, there exists a true sentence that does not have it, or a false sentence that does."

But then again, it still needs to be a property of logic sentences. For example, a property of natural numbers can apply to all natural numbers. — Tarskian

If you refer to "an universal statement that ought to apply to everything", I would agree (assuming I understood your point).

Provability, if I have understood it correctly, means that a truth of a statement/conjecture can be derived from some axiomatic system or logical rules.

With diagonalization, we get only an indirect proof. Here the proof lies on a contradiction that if the statement/conjecture would be false, then we would have a contradiction. Yet here we lose a lot from the direct proof as there's no means to grasp similarly information about statement/conjecture as in the direct proof. That's why the "true, but unprovable" statements have been such a mystery, because we want to have more information about them as we have a direct proof. And of course, people haven't been interested to find "true, but unprovable" statements. Hopefully it's changing now.

So one hypothesis would be this:

Is diagonalization a way to find mathematical statements that cannot be proven by a direct proof, but only can shown to be true by reductio ad absurdum?

And the next, even more outrageous hypothesis:

Is then this also a limit that we can compute and give a direct proof?

Let's just think what our current definition is on what is computable: the Church-Turing thesis. It states that what is computable is what a Turing Machine can compute. What the Turing Machine cannot compute is found exactly by using diagonalization (or negative self-reference) that we are talking in the first place.

But not only is this a informal definition, it is also only a thesis. Meaning literally something that we want to prove. And here we find again an issue where we want mathematics to be something else that it is, if we want to make a direct proof about it, ie a theorem of the Church-Turing thesis. This isn't possible in my view because we are talking about the limits of what is computable or directly provable and what is not.

So what are we missing here?

Basically a proof that defines what is both computable and directly provable in mathematics and what isn't. Because this proof also states what isn't computable and not directly provable, to be consistent with itself, this part itself cannot directly provable, but only be an indirect proof.

My five cents on the issue: The diagonalization itself here holds the key. It could solve a lot of the confusion that mathematics has now.

Love to hear your comment @Tarskian and others too. And if I made a mistake somewhere, please tell, I'm not an mathematician/logician, so I an ad hominem attack on my credibility.

Your statement is a bit different from mine.

Especially to my statements, yes! But that's why you are here, to correct if there's something wrong or is mistaken. :wink:

I fear the dogs will starve to death. — Ludwig V

Fear not, the dogs too are imaginary. And yes, it's a story I invented.

And for Zeno's two dogs, later people (now mathematicians) have put them on a bit different diet. One (guess which one!) has for as food an axiom and the other happily gets it's food in either the surreal or hyperreal system. But they are not with the other dogs that stem from Plato's picked up dog. The question is if they would like to be with the other dogs. Could they be?

So there not dying from starvation. So it's kind of a happy ending?

I agree

This is, in fact, the only hard part in Gödel's proof. The proof for the lemma is very short but it is widely considered to be incomprehensible: — Tarskian

Gödel didn't make it easy. In my opinion Cantor's diagonalization is an easier model. Or basically just use negative self reference with avoiding a Cretan liar situation.

What I don't get is just how little interest the diagonalization (or negative self-reference) gets. Yet with using it Cantor showed that the reals cannot be put into a one-to-one correspondence with the natural numbers. And Turing used it in the Halting Problem and Gödel in the incompleteness Theorems. And here's the key: if we disregard this, we end up in a paradox.

For example, I can write (if I do write it correctly) the following self-referential statement, which is true:

"I can write anything what I write" meaning, that I have no limitations on what I write and what I write is then defined to be something that I wrote or, my writings.

Then let's turn into a negative self-reference, which is also true: "I cannot write anything what I don't write". OMG! Can I write anything? Obviously I can. Does this somehow limit what I can write? No, but it shows that obviously there also is something that I don't write, these writings exist.

Now here's the tricky part: If I make the false assumption that "I can write anything" means that there cannot be anything that I cannot write (if we skip the physical limitations and stick to the theoretical) what would that imply?,

I would have to write also what I don't write, which cannot be.

So in a way, negative self reference in my opinion is a very essential building block for logic. And everytime when someone makes an universal statement that ought to apply to everything, watch out!

Hilbert believed it so strongly that he insisted that all his colleagues should work on proving the above. A lot of people still believe it. You can give them proof that it is absolutely impossible, but they simply don't care about that. They will just keep going as if nothing happened. You can't wake a person who is pretending to be asleep. — Tarskian

The basic problem is that people simply have these ideas what mathematics should be like and don't notice that their own premises, which they hold as axioms (obviously! What else they could they be?), aren't actually true. And when those "axioms" aren't true, we end up somewhere in a paradox.

Easiest misunderstanding to understand was the idea of all numbers being rational. Why? Because math had to be perfect! And then when obviously there were irrational numbers, the story goes that the man, Hippasus, who found irrational numbers was ostracized and when he drowned at sea, it was the "punishment of the Gods". So that at least show how some Greeks thought about it. Yet since some irrational numbers were so useful, irrational numbers were accepted.

Then there's the mess that Russell found out and the collective panic attack that only subsided with ZF-logic simply banning the paradox. There's obviously still a lot of confusion. But we can look at this in a very positive light: there's a lot for us to discover still!

That is the obvious standard line, been for thousands of years. But it's a limitation, when you start from Plato's dog.

Yet doesn't the dog that eats more than any other dog define it different from all other dogs? No matter if there's an Apeiron (endless amount) of dogs that eat less.

The world of mathematical truth does not look like most people believe it does. It is not orderly. It is fundamentally unpredictable. It is highly chaotic. — Tarskian

Perhaps some can see this as chaotic, but math itself is quite logical and hence quite orderly. Unprovability or uncomputability doesn't mean chaotic. Math is orderly, we just have limitations on what to compute or prove.

Of course it matters just how we define Chaos. If it's logical, it surely can be also mathematical.

The fact that people have a difficult time is to grasp that mathematics can be uncomputable (and unprovable). Non-computable mathematics sounds like an oxymoron, right? Wrong, only part of mathematics is computable. Or countable or provable.

Let's take a simple example just how easily we can get true, but not known mathematical entity. Assume a, b and c are distinct numbers that belong to the Natural numbers.

Let's have the equation

a + b = c

if we know two of them, we know the third one. So if a is 2 and b is 3, then c has to be 5. The equation, which is a bijective function, is obvious and easy.

It isn't so obvious when we have an inequation, which isn't a bijection:

a + b < c

If a is 2 and b is 3, then c has to be something bigger than 5 and when c belonged to the natural numbers, it's then 6 or larger. And that's it! Even if it obviously c is a natural number and has a precise point on the number line, not some range, we cannot prove c exactly. The only equation or bijection that we can do is that c=c (and c is 6 or a higher natural number).

The problem rises because we just assume that everything in math has to be provable. And the real culprit here is that when mathematics has risen from the need to count, we have put counting/computing as the basis of all math. That is an error, because we have non-computable math, and hence if we want mathematics to be consistent and logical, somethings got to give.

When, exactly can someone who is capable of being free be said to be coerced? — Ludwig V

Coercion usually means forcing someone to do something he or she doesn't want to do.

As the sun rises over the horizon, is it appropriate to say that the air is coerced to become hotter? If determinism excludes the possibility of freedom, as it seems to, then it also excludes the possibility of coercion. — Ludwig V

That kind of idea of determinism does away with lot of things. Anyway, if we want to hear some who thinks that with determinism there's no free will, then there's for example Susan Hossenfelder:



What is noteworthy is that she talks about emergent properties and decoupling of scales. At 6:40 she describes that the underlying laws still exist, and then at 7:20 even refers to some articles that refer to the Halting Problem, but as they talk about infinates, she disregards them. She also admits that the majority of philosophers believe in compatibilism. Finally on 12:20 she explains why she believes that determinism eliminates free will.

I would argue that this is one usual way people that hold physics to be this all encompassing all answering field, which obviously has all the correct theorems at hand, just look at the world. Great that they are enthusiastic of their field of study. Yet they disregard then the "more is different" argument, the emergent properties, and make simply a category error. The idea that everything is physics is reductionism at worst, when people assume that ontological questions can only be discussed in the realm of physics and through it's models. Because matter is made of quarks etc.

Good antidote would be to understand that the limitation on the predictive ability of LD is mathematics and logic.

But the billiard balls do not roll as they do because LD predicted how they would. — Ludwig V

Yes.
But even you or I could make good predictions about billiard balls at least on a billiard table. LD would easily.

What is crucial to remember in the LD example what Simon Laplace gets wrong is the every part: that LD can forecast everything. In many occasion giving a prediction doesn't affect what is predicted. That the Earth revolves around the Sun even a hundred years from now is a sound prediction. Giving that doesn't effect the future, the Earth or the Sun.

You can even innovate, do really something that hasn't been there before in your mind. — ssu

I'm not at all sure this is relevant for our problem. In the first place, the billiard balls can travel along paths they have never travelled before. In the second place, if we are only free when we innovate, then we are in chains for most of our lives. — Ludwig V

Then I have to remind about the problem that LD had in predicting the future. I don't think LD has any problem in predicting billiard balls as they follow exceptionally well even Newtonian physics. Yet LD has a problem of making an equation when the future depends on his equation, especially the negation of it.

The negative self-reference that can be seen in Cantor's diagonalization has in my view profound consequences. Remember that this negative self-reference is in both Turing's Halting Problem and Gödel's Incompleteness Theorems. It's been long argued starting from J.R. Lucas (1961) and then continued with Penrose that human mind is different because we can understand Gödel's incompleteness theorems and computers cannot, but that argument is a confusing. (Anyway, I think it's nearly pointless to refer to this extremely important theorems because somewhere a logician will jump out and declare that's not what Gödel meant and go through the tedious two theorems until everybody has forgotten what the debate about.)

We do have the limitations that LD has too, yet obviously we can do something that we haven't done, which is hard for a computer computing equations. We can do a lot more than equations, like inequations and simply throw reasonable sounding wild guesses, even.

Are we in chains for most of our lives? Hopefully not literally. But naturally a lot is predictable in our behaviour, yet this shouldn't be a way to denigrate us. Other species are predictable too. What's the problem in being predictable? Large societies need predictability, for example when driving in traffic, I think everybody is happy if you predictably stay on your own lane.

For the sake of argument, if one imagines a human mind as a "decision-making machine", then the freedom of the will is "free to act on the desires and decisions of that machine", in particular as opposed to "forced to act on the desires and decisions of other machines. — flannel jesus

Yes, but in order to be "free to act on the desires and decisions of that machine", which is yourself, you have to have the awareness that you are making a choice / decision. Awareness, consciousness, subjectivity are essential to understand free will.

I see it the other way around. If choices are made because of the physical interactions of all the constituent parts of the brain (whether considered at the level of particles, atoms, molecules, cells, neurons, brain areas, or whatever), due to the properties and laws of physics, and no choice could ever have been/be other than it was/will be, then what is the definition of Free Will that allows for choices to be made freely? Free from what? Other than our awareness of the whole thing, which a boulder lacks, in what way is a path taken for such causes by a person who comes to an intersection different from a path taken by a boulder rolling down a mountain? — Patterner

.
Umm... you answered the question yourself: our awareness of the whole thing. That's it.

Easiest way is simply compare yourself to a computer.

Ask a computer to do something else that isn't in it's program. As a computer just computes. calculates, follows orders and as such, it cannot do this. "Do something else" is a total impossibility, that you can only divert by programming to the computer what to do when asked that. However, the problem doesn't go away. Now before someone argues that we too can't know what our "meta-program" controlling our judgement is, let's think of it another way:

Ask yourself, if you have ever learned something.

If @Patterner of 2024 thinks even a bit differently than the @Patterner that was half of the age as now in 2024. So in the later half of your life, have you have learnt and do differently now, or anything that you think about differently? As the answer is extremely likely yes, then let's go forward.

Now because there was the younger you of the past, who thought or acted at least slightly differently than now, obviously your thinking has changed. You can describe this and tell it, you understand it, you totally can think about these kind of things that have changed subtly along the way. And this is the crucial part: If you are asked "Do something else", you can think what you have done earlier and then really do something else that you haven't done. You can even innovate, do really something that hasn't been there before in your mind. And here it's absolutely no coincidence that this subjective decision making and subjectivity itself poses such a problem, for instance why in neuro-science we have in the hard problem of consciousness. You just said awareness, but it could be described too by consciousness.

Does this refute determinism? Nope. But for entities that are conscious and sentient, free will is a really great model to use!

Dear, oh, dear. I thought it was the causal determinist who was guilty of scientism. — Ludwig V

No no no! Sorry, I wrote badly. I didn't mean you, I meant in general "Now if you go" referring to people who go for scientism. And I'll change it to be more readable! :yikes:

But I really don't think that my "thinking, reasoning and experience " is particularly amenable to the scientific method (methods, approach). — Ludwig V

Thinking and reasoning itself actually isn't so much about using the scientific method. The scientific method is really just a bigger method: it's about how you try to experiment or prove your thinking/hypothesis. Of course sometimes people can be in the happy situation that while studying something else or doing an experiment, they just stumble something they have no clue what it is and where it came from. That's then notice to try to figure what it is, or have you simply made an error.

Now if you go (not meaning @Ludwig V himself) too far with applying the scientific method, the you can hold a bit extreme views of scientism. Usually comes from that the person doesn't understand that other fields do use logic too.

Without basic beliefs, reason is not possible.

Therefore, there is no such sharp distinction between reason and faith. — Tarskian

Yet when you reason, you can change your beliefs. Naturally we do start from our premises, the things we assume to be true. But if by reasoning we come to the conclusion that our starting assumptions were wrong, we change them.

With something like faith, and to love something, that's not so.

Can anyone prove a god, I enjoy debates and wish to see the arguments posed in favour of the existence of a god. — CallMeDirac

Ok,

Even if I'm late to this discussion and haven't looked it through, here's my five cents:

If the difference between faith and reason isn't obvious to people, I urge people first to go and read their actual field manuals here: if you are Christian, read the Bible, if you are a Muslim, read the Quran or if you are a Jew, read the Torah. Now, do any of these Holy Scriptures insist and demand that in order for to find God you just have "really think it through" or "reason it out"? That you'll find God if you just reason enough and think about it? Or is it about faith, something like "taking Jesus to your heart" as in the Christian manual? Fun fact, the difference between the expressions of taking something into your heart or it being an issue of the heart or using your brain is quite old.

And since part of us are interested in the Abrahamic religions here (that I admit, I barely know), they don't actually like worshipping idols. Now ask yourself, if there would be a "proof" of God, what need there would be for the Bible or the Torah or the Quran? You have this proof! Here's the proof, there's God, and that's it!

So wouldn't the this proof be an Idol?

In my opinion, It sure would be. So those trying to prove God are trying to build Idols.

The basics, which I confess I though were universally known are that actions performed by people require for their explanation purposes, values and reasons. Values cannot be recognized in standard scientific determinism, because of the fact/value distinction. — Ludwig V

I'm not so sure about this in a time when algorithms rule our lives and data mining and big data is extremely popular.

Reasons are easily confused with causes, but they justify actions by linking the aim, purpose, ambition, goal or target of action to what is to be done (this is sometimes referred to as the practical syllogism) which is a form of explanation that has no role in causal determinism. — Ludwig V

Well I can easily confuse reason with causes.

In fact, when doing a quick search on the definitions of reason and cause, I got:

reason: a cause, explanation, or justification for an action or event.

cause: a reason for an action or condition : motive, something that brings about an effect or a result, a person or thing that is the occasion of an action or state

When you say that practical syllogism has no role in causal determinism, I think you mean that self-cause is something that causal determinists avoid. But this is a bit of chicken and an egg: the causal determinist will simply say that a person, thanks to his thinking, reasoning and experience came to this conclusion because of the current situation that was can be traced to the past occurences, which can be then traced back to, well, the Big Bang. That there's a practical syllogism there doesn't in my view make the determined determinist change his or her dogma.

The "logical limitations" can be observed in physical Phase Transitions, where a stable organization of molecules can suddenly transform from one structural state (water) to another (ice), but scientists can't follow the steps in between. — Gnomon

The logical limitations start from what we can calculate and prove. What you are describing is more physical limitations that we notice in our empirical tests.

The logical part here is of course when a measurement effects what is being measured. This is something that isn't at all trivial. And then there's things that you simply cannot model in a laboratory.

There's been a lot of discussion of that possibility, but I haven't seen anything that really resolves the differences between them. — Ludwig V

I think it would be productive for this thread if either you or anyone gives the most compelling case just why they cannot be both at the same time. Even if one doesn't personally agree with the argument.

There's either freedom in the gaps or reduction of freedom to causality. — Ludwig V

Can you be more specific what this means?

Determinism is an unprovable metaphysical belief, just as FreeWill is. So I freely choose to believe that when I drive my car I am in command, not the laws of nature or sparking neurons. — Gnomon

Many would say then that you believe in Free Will. But anyway, I agree on the unprovability of these metaphysical beliefs. The thing is that what we can rigorously prove is quite limited. However if and when we make models about reality, why not use them?

In my view both are very useful concepts. I will argue that you can have determinism and free will. Free will is a great concept to use as it easy describing various events and phenomena extremely well. Yet so is determinism too. What we have is logical limitations in understanding a deterministic reality, making predictions about it or calculating what will happen. Additionally people tend to overemphasize in the reductionist the basis, as if on a "lower" level (physics, quantum theory, quarks etc) are more important and profound than other subjects.

In my view the university/academia is a steamroller that if it doesn't crush your innovation and ability to ask naive questions, you are made of steel and can face the world. Coming out of system is good. You should be proud if if the system doesn't crush you. I got a Master's thesis in economic history, but saw the writing on the wall when attempting a doctorate. Never did it. Writing in this forum work far better: the anonymity here makes us equal, and people aren't doing this for work or competing for positions. There aren't too many people here.

By enlarging the academia you simply make all the negative aspects of people working in huge groups more apparent. Fist and foremost, you cannot have a constructive interaction with thousands or tens of thousands of people. Herding, negative affects of group think, bureaucracy, everything that happens with large groups makes it's far less productive. However innovative and exciting Universities try to be, just because of the sheer numbers people they are falling down into something that they were in the Middle Ages. Perhaps it's not so bad, but anyway. Haven't been there for decades now.

Well, this is a philosophy site, so people here do understand why in the university math is studied, even if the applications to engineering etc. are different.

Universities that focus on technology and engineering would then be the places you would refer?

What problem does the math graduate intend to solve except for teaching math? — Tarskian

You could generalize to a lot of what is taught and studied in universities here. Not only math.

Perhaps the real problem is that vocational education is so deprecated and doesn't drawn in the kind of people it should and doesn't go to the level it should.

This begs the question: why does it matter what we think about probabilities? — Igitur

The idea is that there are more unaccounted possibilities in either a category that is similarly rare, has the same effect, or cause the same reaction. — Igitur

Don't forget that people don't simply don't understand probabilities. Even if they know that the Casino always wins, people like to gamble. And how many understand the Monty Hall -problem the first time they hear it, especially if they are made to play the game without any knowledge of the famous example? To understand the connection of information to probabilities is hard, actually.

For example, how improbable it is to win in a lottery can be blurred from how many people do win the lottery. Here in Finland a lot of people play the local lottery, in which you have to get seven numbers correct from 40. To get seven correct has a probability of 1 to 18,6 million. But nearly every week or so someone wins it (even if there's just 5 million plus Finns) and typically only few weeks go without nobody getting seven correct.

Yes, I hear you. One of the basic issues I have with determinism is understanding why people equate it with being forced to do things. — Ludwig V

Have you thought about the possibility of them not understanding the issues at hand well and having misconceptions?

Strict idealism, empiricism also lead to silly generalizations and wrong conclusions. — Ludwig V

Also, yes.

On the face of it, I would have thought that the empirical sciences are more likely to be useful than philosophy. — Ludwig V

That's the magic word: useful.

Yet that aside, ontological questions are important. To understand that the empirical sciences have a philosophical and hence also metaphysical foundation is extremely important. When these foundations are understood, you also understand the weaknesses and limitations of the empirical science, where it can get tangled up in basically ...nonsense. Modern science has huge problems of dealing with Qualia. Add a materialist / physicalist World-view with a staunch belief reductionism, and you will have problems.

Schooling in mathematics spends a lot of time on:

(1) carrying out arithmetical or algebraic procedures that a tool like wolfram alpha can perform automatically.

-> There is no job where you will ever be required to manually carry out procedures that a computer can carry out. — Tarskian

And how can you pick the correct toll, if you don't know the arithmetical and algebraic procedures themselves? By at least learning to do them yourself, you understand them.

Neither activity is meaningful in any shape or fashion. That is, however, what mathematics education is all about. — Tarskian

I disagree. The problem is that there's simply too much math to study at a slow pace. So teachers in school and in the university don't have the time to go really through how some "proof" finally got to be what is now. The pace is so quick it favours memorization and simply those who can use various algorithms quickly.

Aristotle intuitively made a distinction between physical and mental processes in the world. — Gnomon

As we have a lot to thank Aristotle for his ground braking effort to understand the world, I think our scientific understanding has progressed from his time (starting with the scientific method etc). However I do agree that there's a lot we don't understand and have difficulties is grasping the link between the physical and what can be called processes. Strict materialism and physicalism simply leads people to make silly generalizations and to wrong conclusions.

That "separation" was later formalized by others into categories of A> Physics : particular material objects and B> Metaphysics : general mental ideas (universal principles) about those objects. — Gnomon

I didn't know that. I meant metaphysics as things before physics, like the nature of existence (and universal principles) and as the study of mind-independent features of reality. It's really hard to prove something with the scientific method of these kind of basic questions. Hence even if very important, it's not a field you can assume to have dramatic breakthroughs.

Those Generalizations and Categorizations -- "something else" than material/temporal specimens -- are computed by Reason/Logic, which he regarded as a timeless power, capacity or force, accessible to philosophically-inclined humans. For non-rational animals though, there may be only observed things, and no inferred species of things. So, yes, for those who seek holistic Principles instead of isolated Instances, there has to be a separation. :smile: — Gnomon

Well, I think that animals are also rational, so they don't have to be just "philosophically inclined" to have rational thoughts. That we just have and advance language and even the abiltiy to store it (written language) makes us quite different in my view, but still we are animals (even if smart ones).

For example, some mammal living in a herd in the Serengeti might have a mathematical system of counting as "no predators, one predator, two predators, three predators, many predators". It is totally rational and can be totally satisfactory for the animals. If there's more than three predators lurking around the around the herd, it's "Stampede time!" and the time to get the hell out of Dogde. Doesn't matter how much more there are than three, it's far too many. Yet if there's just three, one has to be able to count them: if suddenly there's just two, then one can be lurking in ambush behind you preparing to chew your ass off. So for example the ability to count things is important.

Now we can argue that a math system of counting of "zero, one, two, three, many" is illogical, because why stop there (what happened to four, five, six)? But for a mammal that eats, mates and tries to avoid predators a more advanced system with irrational numbers and imaginary numbers is useless.

But now I went away from the topic.
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I don't quite understand this. I could understand if you were talking about hypotheses. The journey from hypothesis (possibility) to theory (proven) is a long and tortuous one - blurred, if you like. But a model doesn't have a similar journey - unless there is a way in which a hypothesis can be a model or vice versa. Is that your point? — Ludwig V

Well, models can be for example simplified. In economics we can make the premis of ceteris paribus, all other things being similar, and then assume to model something from the economy. In reality hardly anything stays the same and our ceteris paribus -argument wouldn't be valid, if we were really making a model of everything in the economy. Economical models typically try to model a certain part of the whole economy or a certain phenomenon.

Are you saying that any theory that is incompatible with freedom (free will) is false on that ground alone? That's a good start. But many people speak as if determinism was true and we have to bear the consequences, yet seem to believe that determinism is an empirical claim. Even when there's empirical evidence against it, they don't give up on it. I think it has to be classified along with hinge and grammatical propositions, perhaps as a research programme. — Ludwig V

What I'm trying to say that there being a certain future simply doesn't limit in any way free will. If you respond to my argument here, it's going to be exactly made in one way (of course you can modify and rewrite your answer), but this fact doesn't limit you in any way how you respond to me.

Asking what's Real, as if there could be a single-non-context-dependent answer, is the metaphysical way and goes nowhere. — Ludwig V

Yes. Our questions themselves define just what our answers are. There's no ultimate answer, as there is no ultimate question. (Or it's 42, as in the Hitchhiker's guide to the Galaxy.)

Here's my opinion.

Decision-making is a mental process, but mental processing is fundamentally a physical process of the central nervous system. — Relativist

Well, everything is basically a physical process in the physical universe. At least in one metaphysical World view.

In fact then when @Gnomon's idea is viewed as an ontological idea, that "Physical actions are indeed constrained by the limiting laws of physics. But meta-physical (mental) choices are not subject to physical laws --- perhaps only the laws of Logic", it can be argued that he is making the argument that there's something else than the physical. But has there to be a separation?

Above all, do we have to fall into the pit of metaphysical discussions that we have no way of solving (and hence no way to climb out from)? There's no ladder there to reason your way out from the pit.

It's false to draw conclusions from a materialist World view that then free will or making decisions doesn't happen / is meaningless. @Ludwig V has to choose the sushi he wants to eat and he really has to make that decision. Metaphysical questions of what reality really is, don't give an answer to this and deterministic world models are quite useless models to use in this place. That in 200 years we are all dead and @Ludwig V behaved exactly the way as he did when next to sushi table is useless information when our friend has to choose what sushi he eats. And similarly the question just where the decision process happened, or was the @Ludwig V at the sushi table -event predetermined right from the Bing Bang is useless. And neither will it be useful if we go with @Gnomon's idea that there's a mental choice for Ludwig to do, which is different from the physical reality.

We use models about reality to get answers to certain questions. Many times, those models aren't declarations of our views on ontological questions. Yet often the models are interpreted as how we think what reality actually is. The difference between reality and a certain model of reality (that answers certain questions about it) is blurred.

War does things to people. Hopefully The Hague also does something. — jorndoe

I think it starts form when you treat your own soldiers as cannon fodder, expendable, that has a psychological effect on them as they know (and naturally do notice) that they are viewed as so. When you cannot oppose this, but you can do whatever towards the enemy and the civilian population, you can then take out your frustrations on these.

It's actually surprising how the Russian army has now when mobilized turned in many ways into the Red Army of WW2.

Yet if are reminded that you can be court-martialed for killing, torturing or raping civilians, then that limits these kinds of actions.

Here's a good interview where David Wolpert goes through his reasoning why Laplace's Demon cannot make his forecasts (starting at 4:16)... among other things closely related and others not so. The name given to the talk below is misleading and can lead to misunderstandings as this isn't at all a religious discussion.

The effects of feedback of predictions on future action is very well known in economics, isn't it? — Ludwig V

Actually as I've studied economics in the university in the 1990's, at least it wasn't so back then. Economics just tries to use dynamical models which don't blow up. Yes, speculative bubbles, self enforcing expectations, Keynesian Beauty Contests are known, but their use is the problem! The math isn't there to use. Economics tries to use mathematical models. And, well, I think you can guess the problem lies (as we have been talking about a limitation on mathematic modelling).

(Sorry, I misspelled the name, it was Rukavicka.)

I'm not quite sure what you mean. — Ludwig V

The limitation on the modelling isn't generally known, although some discussion about it has been said.

Take for example economics: in the 1930's two later-to-become Nobel economists had a debate in an American economic journal where the other pointed out this basic problem in forecasting (if forecast itself effects the future). The other one simply attempted to refute this by stating there simply has to be a correct model, because there is an outcome. He just stated that perhaps in the future we will know how to do this model. Well, this didn't actually answer the question, but economics simply attends those kind of events where forecasting can work (mathematically).

And secondly, the result here isn't generally accepted or public knowledge. Just look at the references, videos or writings about LD. The usual idea is that since we have quantum physics, LD isn't happening because the physics isn't all Newtonian. But that's it.

I copied the abstract above (so hopefully the magazine won't sue me :yikes:). So, it's basically what we were talking about. If you look at how Turing made the argument in his Turings Machine, you would notice that it's also similar to this. And of course, Gödel's Incompleteness Theorem's are even more difficult.

Here's what in 2014 Josef Rukavicka wrote in The American Mathematical Monthly Volume 121, 2014 Issue 6, which goes total the same lines as we have discussed:

Rejection of Laplace’s Demon

In the 19th century, Laplace claimed that it might be possible to predict the future under the condition that the positions and speeds of all items in the universe at a certain moment were known [3]. The entity that is able to make such a prediction is often called Laplace’s demon. This topic has been extensively discussed and investigated (see, for example, Hawking’s lecture [1]).

Recently, Wolpert [4] defined ”inference devices” and proved several theorems associated with them. One of the consequences of the theorems is that he disproved any possible existence of Laplace’s demon. The proof he used is based on Cantor’s diagonal argument. In this note, we present a much simpler proof using the Halting problem of a Turing machine [2]. Recall that the Halting problem can be stated as follows, ”Given the description of a Turing machine with some input string, in general, we cannot determine (predict) whether that Turing machine will halt. ”

We claim that if it is impossible to predict whether a Turing machine will halt, then it is impossible to predict the future. In other words, if we claim that we can predict the future, then we must be able to predict whether a Turing machine will halt. This result can be easily presented without any reference to Turing machines, or even to mathematics at all. Suppose that there is a device that can predict the future. Ask that device what you will do in the evening. Without loss of generality, consider that there are only two options: (1) watch TV or (2) listen to the radio. After the device gives a response, for example, (1) watch TV, you instead listen to the radio on purpose. The device would, therefore, be wrong. No matter what the devicesays, we are free to choose the other option. This implies that Laplace’s demon cannot exist.

It's the same thing we have been talking about here, and here's the catch:

I think it's too easy, too obvious when you stated as above (once people get it), and people don't understand how important the whole issue is. People will go just: "Huh? Well, Laplace's ancient history, anyway". The fact that this really is important in math doesn't come up. Or hasn't yet as do note that the timeline here: Wolpert made his papers 2000-2008 and Ruckavicka stated the simplified version in 2014.

Why I'm so obsessed about this (one could argue)?

The limitation is essential part of logic, yet it's not understood as to be so. And I think it actually defines a part of mathematics that hasn't been accurately describe: the non-computable. How does mathematics even have a field that is non-computable and the link of the non-computable to objectivity and subjectivity isn't made clear either.