I think everyone cares when they are on that operating table. They sure hope that there is a right way to transplant a heart, for instance. — mrcoffee

Someways definitely better than others but people still die. In fact, accidental deaths in hospitals are one of the leading causes of deaths in the U.S. one study putting it at over 400,000 per year, the third leading cause of accidental death.

Indeed. We just generally try to maximize our chances for success.

Yes I agree that logic is impossible without this fundamental first step. But if contradiction is inherent within the first step, don't you see this as a problem? — Metaphysician Undercover

I'm not sure that we have a problem. We can distinguish between the individual marks and then categorize these marks. For instance, let A = {s2e, 2rt, e42}. Let f be defined on all strings over lower case letters and decimal digits so that f(s) = 0 if s does not contain the symbol 2 and f(s) = 1 if s does contain the symbol 2. Then f(s2e) = f(2rt) = f(e42) = 1. A less formal example would be three different dogs, each recognized as belonging to the category 'dog.' I can only offer this example because we already understand this category 'dog.' My formal example suggests how math basically scrubs 'ordlang' logic of its ambiguity, which is gets from our fuzzy language, so that it's structure can be focused on and examined.

The first law is the law of identity. — Metaphysician Undercover

I find the 'law' metaphor a little awkward here. "Each thing is the same with itself and different from another." This is a tautology or close to a tautology. We can give the same thing different names, but we wouldn't generally give different things the same name, remembering that a name would not be a fixed context-independent token in ordlang. I can talk about 'John' successfully if the context specifies which John and enriches the token 'John.' Of course a formal system avoids what could go wrong here by insisting that the token completely specify the entity. In a formal system the token is the entity. So we scan the page, see the mark 'abc' which is microscopically different from the mark 'abc' on the previous page, and map both automatically to the token or sequence of tokens 'abc.' (This is slightly tricky, because I can only talk about these marks by using tokens.

The second law is the law of non-contradiction. — Metaphysician Undercover

This 'law' again just seems to codify or tautologize the syntax of 'ordlang.' It's indulgent of me to try and fire up a meme, but I am trying to point at an inexplicit know-how that is also invisible in its smooth functioning.

Aren't we obliged to either forfeit the law of non-contradiction, or go back to our mode of identification and rectify this problem of contradiction inherent within identification? — Metaphysician Undercover

I hope I addressed this above, if obliquely.

I'm no mathematician, but I've noticed that set theory has contradiction inherent within it as well. — Metaphysician Undercover

Depending on what you mean by 'contradiction,' that would make you famous. I think you mean that you have philosophical reservations about set theory ( the formal system itself). I can relate to that. When we talk about systems from the outside, we leave the cosy objectivity available within the system, as I see it. If I understand correctly, a mathematical contradiction in this context would be a 'legally' generated string with certain properties. Or rather it could be put in such a form. In practice almost no one works in 'machine code,' though I understand that to be a speciality itself. For example: https://en.wikipedia.org/wiki/Automated_proof_checking

I do not believe that the resolution to the problem of contradiction being inherent within the first step, is to introduce other contradictions to cover it up. — Metaphysician Undercover

If we view formal systems as pieces of technology, then it makes sense to me that we might tinker with them. Cantor's original set theory was beautiful and revolutionary, but it allowed some contradictions. Mathematicians didn't want to just give up on something so promising, so they added a few constraints in the hope that they could avoid contradictions. They at least fixed the obvious ones, and they did this without losing much. I have Suppes' book, and it follows the history closely, with some great quotes of the philosophical motivations of those who fixed set theory up. The axiom of infinity especially begs for philosophical dicussion: https://en.wikipedia.org/wiki/Axiom_of_infinity.

To demonstrate imperfections within something is to demonstrate that it is less than ideal. It is not necessary to show the ideal, in order to demonstrate that what we have is not ideal. — Metaphysician Undercover

To me this is far from obvious. How do we evaluate/demonstrate the relation 'less than' without an image of the ideal? I will agree that we can prove the impossibility of certain systems. Every finite field has order p^k where p is prime and k>0, so we can't have such a field with 6 elements. If we determine that a finite field with 6 elements would somehow be useful or beautiful, then there would be a gap between the real and the ideal. But if we consider possibility (that the system works) to be an essential feature of the ideal, then I don't currently see how we couldn't immediately institute a particular vision of the ideal.

Admittedly, we don't always know ahead of time whether we are desiring the impossible.

To demonstrate problems within a system does not require that one put forward resolutions to the problems. — Metaphysician Undercover

I agree in general, but my concerns above address the case of formal systems.

I'm not sure that we have a problem. We can distinguish between the individual marks and then categorize these marks. For instance, let A = {s2e, 2rt, e42}. Let f be defined on all strings over lower case letters and decimal digits so that f(s) = 0 if s does not contain the symbol 2 and f(s) = 1 if s does contain the symbol 2. Then f(s2e) = f(2rt) = f(e42) = 1. A less formal example would be three different dogs, each recognized as belonging to the category 'dog.' I can only offer this example because we already understand this category 'dog.' My formal example suggests how math basically scrubs 'ordlang' logic of its ambiguity, which is gets from our fuzzy language, so that it's structure can be focused on and examined. — mrcoffee

As I said, my mathematics is not good, so I don't see how the example deals with the problem. Say object #1 and object #2 are seen to be different, but not identified as different. They are both identified as "dog". We know the objects are distinct, be we are identifying them as the same. You might say that this is just a categorization, but for the sake of the logical process which follows the identification, they are the same. So for the sake of the logical process they are said to be the same, when they are really different.

To me this is far from obvious. How do we evaluate/demonstrate the relation 'less than' without an image of the ideal? I will agree that we can prove the impossibility of certain systems. — mrcoffee

The meaning of "less than" is not demonstrated, it is stipulated by definition, in reference to an order. I don't think "less than" can be judged without reference to the definition, and therefore the order. If you want to argue that a definition is an ideal, I don't think you could succeed because definitions are not perfect, due to the ambiguity of words.

And, since unities can be subtracted from and divided, as well as added to and multiplied, I don't know how you would designate the ideal order. To have an ideal order, I think would require having an indivisible first unity. Zero might serve the purpose, but it allows for the possibility of a negative as well as a positive order. They cannot both be "the ideal order" unless "zero" is defined in relation to some ideal good. Then we'd have less and more in relation to that good.

But if we consider possibility (that the system works) to be an essential feature of the ideal, then I don't currently see how we couldn't immediately institute a particular vision of the ideal. — mrcoffee

I think that this is contradictory. "Possibility" implies necessarily a multitude. There cannot be just one possibility or else that possibility would be the only possibility, and therefore a necessity and not a possibility. This contradicts "a particular vision of the ideal", which implies necessarily the one and only. In other words, it contradicts the notion of "the ideal" to allow that possibility inheres within.

Say object #1 and object #2 are seen to be different, but not identified as different. They are both identified as "dog". We know the objects are distinct, be we are identifying them as the same. You might say that this is just a categorization, but for the sake of the logical process which follows the identification, they are the same. So for the sake of the logical process they are said to be the same, when they are really different. — Metaphysician Undercover

Right. But in the science of formal systems we discover the relationships of categories/symbols/tokens and not of the marks we need to aid memory and communication. The theorems aren't about the marks. Beyond that, there's no denial that the marks are different. There's just no interest in the mark except as the representation of a category.

If I draw the letter a in two ways, even a child can agree that the marks are different and yet the 'same' (the same letter). This is an informal computation of the many-to-one function from marks to symbols. Some might prefer to use 'symbols' for what I mean by marks, which is fine. So for clarity I can just talk about the categorization function or categorization itself, which is allowed to place 2 or more different objects in the same conceptual bin. This is just an ability we find ourselves with. Existence is ultimately mysterious, etc. But I don't think there's problem with categorization. A person would have to use categories successfully in order to argue for their failure.

The meaning of "less than" is not demonstrated, it is stipulated by definition, in reference to an order. I don't think "less than" can be judged without reference to the definition, and therefore the order. If you want to argue that a definition is an ideal, I don't think you could succeed because definitions are not perfect, due to the ambiguity of words. — Metaphysician Undercover

But surely you didn't mean the usual order on the integers or real numbers? My point is that if something is less than ideal within or about mathematics, that this would tend to involve a notion of the ideal. Anything ideal for mathematics would already be in the right form for immediate adoption. I did give an example we could want a particular formal system (a field with 6 elements) and discover that such a thing is impossible. If we want the impossible, then math (in this case 'logical reality') is not ideal.

Yep, none of our concepts are linguistically representable, because our intended use of rules cannot be finitely represented in terms of rules and signs. We can at most express what we mean and intend, but we cannot reduce meaning and intention to rules and signs.

This critical anti-realist insight renders rules and laws as having trivial epistemological significance, since the meaning of rules and signs is ultimately grounded, explained and justified in terms of our behavioural dispositions, as opposed to our behavioural dispositions being grounded or justified in terms of mind-independent rules and laws. This in turn ought to lead to a rejection of the free-will-determinism dichotomy.

Right. But in the science of formal systems we discover the relationships of categories/symbols/tokens and not of the marks we need to aid memory and communication. The theorems aren't about the marks. Beyond that, there's no denial that the marks are different. There's just no interest in the mark except as the representation of a category. — mrcoffee

This is not "the science of formal systems", this is philosophy. In philosophy we are concerned with understanding reality as a whole, so we cannot dismiss certain contradictions and inconsistencies as irrelevant to the field of study.

If it is necessary that we take two distinct things, which have a very similar physical appearance (two distinct instances of a symbol), and assume that they are "the same", despite the fact that they are clearly not the same, in order to understand some aspect of reality, then as philosophers we ought to recognize and take interest in this, to determine what the implications of such a contradiction might be.

If I draw the letter a in two ways, even a child can agree that the marks are different and yet the 'same' (the same letter). This is an informal computation of the many-to-one function from marks to symbols. Some might prefer to use 'symbols' for what I mean by marks, which is fine. So for clarity I can just talk about the categorization function or categorization itself, which is allowed to place 2 or more different objects in the same conceptual bin. This is just an ability we find ourselves with. Existence is ultimately mysterious, etc. But I don't think there's problem with categorization. A person would have to use categories successfully in order to argue for their failure. — mrcoffee

The point is to recognize the difference between "the same" and "similar". We all agree that there is a fundamental difference between these two. I would say that it is a categorical difference, "the same" always indicates one and only one, the one and only, while "similar" always indicates a multiplicity. The law of identity, as expressed by Aristotle claims that "the same" refers to the one and only. This is supported by the Leibniz principle of indiscernibles, if it appears as two distinct things, but the two are absolutely identical, then they are in fact the same, one not two.

I agree that we can categorize, and place similar things in the same category. Where I disagree is that it is proper to call these similar things "the same". The category itself is "the same", as the one and only, but the individual items are not the same, they are similar. Therefore we have a categorical separation between the category and the members of the category. The category is "one", the members are "many". If I remember correctly, set theory violates this categorical separation, being based in a fundamental category error.

But surely you didn't mean the usual order on the integers or real numbers? My point is that if something is less than ideal within or about mathematics, that this would tend to involve a notion of the ideal. — mrcoffee

Yes, we refer to the notion of "the ideal". Let's say that it must be perfect, cannot be otherwise without loosing the status of "the ideal", therefore it is unique, the one and only, the ideal. If it can be demonstrated that any particular principle does not fulfill this criteria, then we can conclude that the principle is not the ideal. I agree that in excluding the principle from the category of "ideal", we refer to "a notion of the ideal". Do you recognize the difference between "the ideal", and "the notion of the ideal"? We make a category, "the ideal", and have a notion of what is required of something to be put in that category. But the category may be empty, like the empty set. There may be nothing to put in that category. There is a notion of the ideal, but we haven't found the ideal. That we have a notion of the ideal doesn't mean that we have the ideal.

This is why I used zero as the principle for ordering. Let's say someone claims that zero fulfills our notion of the ideal. The argument is that we haven't found any ideal, the category is an empty set, therefore zero is the ideal. However, zero allows for the possibility of ordering toward the negative or the positive, two distinct possibilities. So there is inherent within "zero" two distinct possibilities. Therefore it cannot be the ideal because the ideal must be one unique perfection. The ideal is like the empty set, but it cannot even be represented as zero, because we cannot put zero into that set, because this leaves it not empty.

This is not "the science of formal systems", this is philosophy. — Metaphysician Undercover

And it's dark at night.

In philosophy we are concerned with understanding reality as a whole, so we cannot dismiss certain contradictions and inconsistencies as irrelevant to the field of study. — Metaphysician Undercover

Unless we dismiss them at dead ends or as not really being contradictions. We decide all the time (implicitly at least) what is and is not worth talking about.

If it is necessary that we take two distinct things, which have a very similar physical appearance (two distinct instances of a symbol), and assume that they are "the same", despite the fact that they are clearly not the same, in order to understand some aspect of reality, then as philosophers we ought to recognize and take interest in this, to determine what the implications of such a contradiction might be. — Metaphysician Undercover

I don't see any mystery in the process. I know very well that two marks are different as marks. I just want these marks to function as symbols. The formal system example just shows this categorization at its nakedest. As you read this sentence, you see words and not marks. As you write your replies to me, you think in terms of words/symbols. This know-how is ultimately mysterious and elusive. I can't say what it is to mean in some conclusive end-of-conversation way. I think it's interesting to try, at least for awhile. But then I may want to actually look into something that I can be far more conclusive about. I think this touches on how virtuous an individual finds philosophical hand-wringing.

Now my use of the phrase is already taking a certain side. I'm an ex-philospoher being a smart-ass, one might say. But I think it's good 'philosophy' to demystify certain classic poses. My 'complaint' would be perhaps that this 'handwringing' is too easy. I think that's why the genre isn't admired much by its non-participants. As a reader of philosophy, I certainly don't side with 'anti-intellectualism,' but sometimes 'anti-intellectualism' is just the flip side of handwringing --a word the handwringers have for those unimpressed by their sweaty palms. To be clear, I'm interested in a synthesis of both sides, and I try to sort effective handwringing (which is perhaps the most powerful kind of talk and thought) from running around in the same old circles compulsively. I'm well aware that I am far from the objectivity of formal systems in presenting my views and preferences.

Do you recognize the difference between "the ideal", and "the notion of the ideal"? — Metaphysician Undercover

That's tricky in math, though. We can have whatever we dream up, with a certain constraint on the dream that keeps it mathematical. I mentioned a field with 6 elements because I can imagine a real world application that might be possible if such a field were possible. Maybe I'm designing a code and such a thing would be convenient, the perfect size. The practical ideal exists, and yet the logical structure of human cognition makes such a thing impossible. Abstract algebra can be read as implicitly psychological. While proofs can be formally true and meaningless, they tend to be written from and for an intuition of necessity.

This is why I used zero as the principle for ordering. Let's say someone claims that zero fulfills our notion of the ideal. The argument is that we haven't found any ideal, the category is an empty set, therefore zero is the ideal. However, zero allows for the possibility of ordering toward the negative or the positive, two distinct possibilities. So there is inherent within "zero" two distinct possibilities. Therefore it cannot be the ideal because the ideal must be one unique perfection. The ideal is like the empty set, but it cannot even be represented as zero, because we cannot put zero into that set, because this leaves it not empty. — Metaphysician Undercover

I can't understand what you are trying to say here. Since '0' is just one part of a system (or of many systems), I can't imagine anyone saying that it itself is or is not ideal. I can only guess where you are coming from, but I can say that I found math far less metaphysical upon studying it than I first understood it to be. Or rather it's metaphysical in the driest and most desirable of ways. It works with basic structural intuitions.
https://en.wikipedia.org/wiki/Peano_axioms

Unless we dismiss them at dead ends or as not really being contradictions. We decide all the time (implicitly at least) what is and is not worth talking about. — mrcoffee

This is a difference of opinion then. You dismiss these inherent contradictions as "dead ends", "not worth talking about". I consider them as having important ontological significance.

I can't understand what you are trying to say here. Since '0' is just one part of a system (or of many systems), I can't imagine anyone saying that it itself is or is not ideal. I can only guess where you are coming from, but I can say that I found math far less metaphysical upon studying it than I first understood it to be. Or rather it's metaphysical in the driest and most desirable of ways. It works with basic structural intuitions. — mrcoffee

The point is that "0" does not work within a basic structural intuition, as you claim, it works only by arbitrary designations, different intuitions which vary. One cannot say what "0" symbolizes because depending on how it is used, this varies.

For example, put "0" on a number line. You can count down, 3,2,1,0,-1,-2, etc.. Here, "0" is just an equal integer. You count two equal intervals between 3 and 1, and likewise, two equal intervals between 1 and -1. The intervals are all equal. This clearly does not represent what "0" really does in mathematics, it has a special place, of greater significance than any other integer.

In other applications, "0" occupies a special place, unlike any other integer. This is because when we count down the positive integers we are proceeding toward "less than", but as we pass zero we cross a categorical separation, into the negative integers, so the negative numbers increase and we actually count "more than" of a different category, the negative integers, as we continue to count down. Counting down from zero, through the negative integers is actually counting up, increasing the number, of a categorically different thing, the negative rather than the positive.

The different, arbitrary, functions of "0" become evident when you try to multiply negative numbers. One convention says that when you multiply negative numbers, a double negative makes a positive, so negative numbers multiplied together makes a positive number. But this is not a very sensible convention because it doesn't allow that there is a square root of a negative number, and it doesn't reflect the fact that when we count into higher and higher negative numbers, we are actually increasing the quantity of a different category, the negatives, not simply descending by integer. So another convention allows for imaginary numbers. But no existing convention really represents "0" properly, as making a categorical division, such that we increase from zero into two distinct categories, the positive, and the negative.