What does a successful mathematical theory look like after 50 years?

Maybe not those particular places (though I admit to not even checking), but there are plenty of other scholarly outlets:

http://dl.acm.org/citation.cfm?id=291182

http://www.sciencedirect.com/science...20737375800022

https://journals.aps.org/pra/abstrac...RevA.62.052309

http://www.sciencedirect.com/science...75960109005520

More importantly

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

Like game theory looked 20 years ago, for example.

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

So, you don't want to compare with...

Graph theory in 1786: https://en.wikipedia.org/wiki/Graph_theory

The theory of roots of polynomial equations circa 150 BCE (and probably even earlier): https://en.wikipedia.org/wiki/Polynomial#History

The theory of complex numbers 50 years after the first attempted usage (which appears to be sometime around 100 AD): http://rossroessler.tripod.com/

?

I really think you're too rigid (and perhaps shallow) in your perspective. The history of math is full of ideas that existed for a long time before anyone was able to make significant progress with the idea.

Interesting point, but it still seems more reasonable to compare theories from the same epoch, given that they have similar opportunities for dissemination.

I think the history shows us trends of human interest far more than the true depth and success of various mathematical theories. While mathematical theories do depend on the availability of human dissemination, their "success" is dependent more upon what's trendy and fashionable to study, and the availability of various "technologies" (which can include other mathematical theories).

An excellent example of this is the "sudden" interest in machine learning.

http://www.mlplatform.nl/what-is-machine-learning/

It was studied a bit in the 1950s, but then not a lot of work on it was done until the 1990s because suddenly computers were a much more approachable object. Even today, the bulk of the usefulness of machine learning is being developed on rather "simple" ideas, and the success is somewhat dependent upon computational power more than really deep insights into the mathematical mechanisms of machine learning. In other words, we're 60-70 years into the theory, but the theory is still really just barely in its infancy.

Also, see "fuzzy logic" and "artificial intelligence" above.

For lastcardcharlie, what point will you have made about the ontology of logic if we accept your 'pragmatic' classification?

You are saying there are many logics, I am saying there is one logic. If there is one logic, that seems to me to make it something special, or even spiritual, and I am curious as to its nature.

My remark that the vast majority of mathematicians use the same logic was meant as an example, and my argument is more general and naive than that. Take any arena in which arguments are made: nobody thinks to clarify which of your many logics is the correct or appropriate one to use. A judge does not make rulings to that effect, for example. Lawyers might argue well or badly or incorrectly, but they are all using, or supposed to be using, the same logic. Everyone is using the same logic.

So do you deny that the other logics that were put forth actually exist?

You have a statement of the form "if P then Q" and it seems that P has been established to be false. So I'm not sure how much it matters whether this statement is true or false.

There are many situations outside of math in which we use non-binary logics. For example, a jury may find a person not guilty of the crime even though they do not believe the person is innocent.

Logic is still simply rules of truth preservation for a defined language?

I don't see how it's logic that is the arena of spiritualism or mysticism. The idea of logic is a pretty straightforward, operational one, and is entirely agnostic with regard to how anything actually is true. You could imagine a similar set of language and rules about how to manipulate sentences concerning 'has a face' or something else. I don't see how the nature of those sets of rules would be any different to logic, just far less useful.

As Aaron is pointing out whether a language is useful or not doesn't prevent it from having a logic, or from that logic having the same nature as any other logic.

In arguing for one logic, then, you're really arguing for one formal language behind reasoning deemed 'useful'. Whatever spiritual or mystic power you see is surely in the reasons why there's only one useful such language, as even useless languages can have a logic. Its usefulness is probably to do with that language's relation to the world. Possibly your theory of meaning or truth.

It's more a programmer's and engineer's type of logic, allowing you to use continuous variables variables, simplified syntaxes and build control functions that take input data that might lack minute precision.

Of course it isn't "sexy". Translating data into word syntaxes and logic operations that yield guesses instead of "true or false" will never look elegant or "sciencey".

It's also important to note that the underlying principle of fuzzy logic (somewhat unpredictable solutions between 0 and 1) is often used in soft computing. Some methods might have surpassed fuzzy logic in popularity, but fuzzy logic's impact on these can't be denied.

One thing that does make fuzzy logic great is that it is very cheap and cost-effective. It's a nice way of building "fake AIs" in games for example (add a small "scramble" to your "AI's" ability to "see", make it use a fuzzy set, offer a solution). Suddenly your "monsters" are less predictable, but they'll intuitively feel like they have a system to their decisions... all at the cost of some very minor lines of code added and very little extra computing needed.

This technical debate about the subject of logic should be elsewhere. Not only for the obvious reason but also because religious people will erroneously use it to bolster their case.

Logic , as in thinking, has three(3) aspects which occur simultaneously within human thought processes.

The first is a "conclusion" as man is continually "concluding" in his thinking. this may not b evident as the conclusion is expected to be last but if I see a horse this is a "conclusion' within my thinking or perceptive abilities.

The horse "speaks for itself" for at this stage I am affected by the horse without adding my personal proclivities to this "conclusion'.

The next step is a "judgment" as I might glean that "the horse is an animal". With respect to this "judgment" on my part, the horse becomes part of me, so to speak, and in this I experience a "feeling". the horse has entered into my inner nature , so to speak.

My "feelings" about the horse can and will become different from my neighbor's for this "feeling" is specific to myself.

The third aspect of logic comes forth from the deeper aspect of my soul nature and I have a "concept" of the horse. It is here where a person could go astray for the "concept" is within the realm of supersensibility where thinking lies and is the completion of an act of knowledge.

The knowledge is the combination of horse-concept which is the reality everyone seeks. To only deal with the horse gives you half of the world or cosmic reality whereas in the act of bringing concepts with ones sensible percepts the world is made whole.

This little part of the cosmos which the individual man brings together with the sense bound horse and the concept was always there for it is not created by man but hidden in the percept, so to speak again.

Another way of saying this is that one's thinking is not personal for thinking is a "sense organ" to which we all have access. Through thinking I experience myself united with the stream of cosmic existence.

From what I understand "logic" was never meant to,nor does "prove" anything .

Recapitulating ; The horse is a "conclusion".

Bringing the horse into myself I form a "judgment" with its concomitant feeling, my individual feeling .

I then experience the "concept" which is the partial reality the horse, in thinking. This is not "my concept" but the relational exegesis as stated above.

This also leads to the battle between "nominalism" and "realism" where the "nominalist" considers the word "horse" as a "tag" with no inherent reality, thus denying the reality of the "concept".

The "realist" states the "concept horse" is within thinking as stated above and therefore of the most "real". Aquinas debated this with an Islamic scholar whose name eludes me but who is quite famous in the philosophical world, as least the guys on my street corner would agree.

Goethe in his "archetypal phenomenon" reached to the reality of the concept and higher which would be called the "idea" which is of increased substance and life.

Modern science is by and large the path of the nominalist for it denies the supersensible in its active approach to reality.

A nominal approach to a cat wold be if I named the cat "pussy" which is justified within the nominal. The cat(not the word "cat") but the "cat" himself is within the conceptual as part and parcel of the real.

No continuation from me, at least. I doubt my oneness of logic bull**** is defensible, and even if it were, I don't know enough about logic to be the one to defend it.

I'm posting like a dickhead at the minute.

I agree, there's only one logic with language and thought differing in the sense that German, Chinese, and English concepts are the same as the concept stands on its own, while the individual languages can and do present differing pictures of a particular thought.

To repeat , the thing on a German's shoulder is called "kopf" and is related to its roundness while the Italian calls it "testa" as in last will and testament. Language also evolves but the thought, or that of another realm is the same for the German, Italian, Chinese and even the Brits.

Who can fight the multiple referrals of differing logics ; appears to me, to be an obstructionist cover.

By the way, no you're not.

Cheers, Carlo.

These crows are using logic, right?

I think your point is interesting. Let's remember for example that fuzzy logic can be written out in binary logic. If not, it wouldn't work (except as an approximation) in binary computers to begin with.

So, your point about binary logic being superior might very well have merit. I was objecting more to the usage of the term "useless" when I defended fuzzy logic. If we think of it is as a "higher order language" (allowing us to describe groups of binary components / equations with words), we see that it (and similar techniques) can be very useful.

Still, when you reduce things to binary computations you do meet some problems. I don't have a formal logic background and I'm not very well versed in it, so I can't answer it from that perspective. But you do have things like the Church-Turing thesis which are interesting in this discussion.

What prevented me arguing further was reading this:

https://pdfs.semanticscholar.org/9ad...2e713d8e73.pdf

Which is like Pyatnitski's definition, except that that definition ignores syntax/proof theory. My understanding of logic does not stretch to knowing whether this collection of languages immediately blows One Logic out of the water.

Well, as I said my understanding of logic is fairly basic. I'm sure however that there is no agreed upon consensus on exactly what logic is.

To me logic is the calculative element of information or meaning, the other parts being values and syntax. You can't really have information without either, and I'd even question if separating them is possible. I mean, I can say "blueberry" and it carries a value, but there has to be a calculation somewhere that ensures the word can carry that value - and there has to be a syntactic element for us to know how to set up the word. But all these elements are interacting and it doesn't seem intuitive to me that they can do anything without each-other. Or in simpler terms: At any point where we try to reduce the concepts of information or meaning into parts, we have to use information and meaning to describe those parts.

To use an analogy: If I paint a picture, the final picture is the information or meaning. Logic is how the paint behaves on the canvass, syntax is the patterns necessary for the painting to portray something and values is the things we want to paint. But the concept of a painting becomes meaningless without those elements, so we're just in a spiral of endless definitions if we try to figure out exactly what it is.

But at this point I don't even know if I'm making sense or using terms in some manner which contradicts great thinkers.

The applications in artificial intelligence and machine learning are ways to force binary decisions. For example, you may ask it to classify some image. At the end, you need to know whether or not it's an image of X, and that binary output is what you get.

But in the background, the machine learning algorithm may be saying that it's a 51% chance of being X and a 49% chance of not being X. So both logics are in play.

So while it may be "written out" in something that can be interpreted as a binary logic (True/False, standard binary logic gates, etc.), there are ways of reading the conclusion that are not binary

Guys - OP here. I haven't deserted the thread. I will reply soon. I know that my comments were inflammatory, and I need to get my thoughts together and put together a reasoned/reasonable response to a lot of the replies. I apologize for the delay.

I will quickly respond to say "this", though. In the OP, I meant strictly syllogistic logic. All these debates about "what is logic" are irrelevant, inasmuch as it relates to my OP.

I will respond more soon. Again, apologies for the delay.

I wasn't going to reply to most people tonight, because they had some smart things to say, and I need time to think about it. This is just word salad, though, so I think I can safely respond to this.

You did not respond safely. I'm not optimistic that you've made the fundamental intellectual shift that's causing you to get so many things wrong in the first half of the thread.

Don't flatter yourself - you weren't one of the people that I was saying had made smart comments.