"object X lies on plane P", if, and only if, "object X lies on plane P" actually expresses an object X lies on plane P as a mathematical abstract reality, and object X lies on plane P obtains in this mathematical abstract reality...
We have no way to specify what it means for an abstract object to 'lie' on a plane." We do? Chris Langan's "question and answer together". In "math world" we say "2 + 2 = 4" for each case? From a free association perspective; "object " "lie" and "plane" need only meet in a certain order?
I don't know who Chris Langan is? Throw a name at me who is recognized in the field of philosophy, like Hilary Putnam, or Michael Dummett, or Daniel Dennett.
This 'free association' is all based on human language and extends from human imagination. The point about defining truth in terms of abstract concepts is that they don't have any tangible meaning when talking about the actual words 'object', 'lie', 'plane', etc. Without something tangible, we can only define our terms to a certain point, at which we have to work based on some kind of belief system. For example, if someone asked me what does it mean for an apple to lie on the desk, I merely have to take the apple and show them how the apple presses against the desk. Perhaps I can press the apple into the desk such that it starts to mash into the desk, all of which I am trying to convey a tangible meaning and thereby translate 'apple', 'lie', and 'desk' into a set of images and sounds that provide tangible meaning. In the case of abstract objects and rules, we cannot reduce these abstractions to anything other than abstractions. That's okay if we want to believe something that requires an abstraction since many very important concepts of the world fall into this category, but when defining an abstract concept like truth with the hope of giving us a sound definition (remember a definition is a reduction to something more tangible), then we must meet a higher criteria than just axiomizing terms. These terms must take on something we can directly experience and experience in a manner which doesn't allow for inconsistencies. If inconsistencies exist (e.g., everyone has a vague or varying definition of the term 'lie'), then obviously you haven't defined your terms. This is a key impediment to defining mathematical truth since this is what we cannot accomplish. Now, I think it's true that nothing in our world is really fully defined. There's a word trick we can all play where we could define a word and then ask to define that word, etc. At some point we simply run out of words to define the remaining words (i.e., without mentioning words that were already used). Nevertheless, the words that are more common in our usage (e.g., 'he', 'her', 'them', etc) can be further defined by pointing at the actual reference. With that kind of ostensive definition, a picture is worth a thousand words and a better understanding is obtainable. The reason why this is important is because with good definitions (good reductions), we can hopefully eliminate concepts that otherwise might disrupt our understanding. For example, if mathematical truth is some kind of Tarskian 'M2' instead of its truth conditions (e.g., if A then B), then we don't look to understand mathematical truths by their truth conditions. The definition (reduction) avoids looking in the wrong direction.
"Touch" is not necessary; they need only be defined as MEETING within a certain context which might be quite generally defined for math purposes. This allows transferability of concept between more specified geometries.
'Meeting' or 'touching' are English terms that have reference to tangible events that take place in reality. But, we presumably aren't talking about reality, so you have to explain what you mean - otherwise the terms don't have enough meaning to understand the Tarskian quantifiers that we are placing as truth quantifiers. What happens if someone says "excuse me, the term 'meeting' is a symbolic word that can be reduced to an actual meeting in the real world, that's how we can understand what a 'meeting' is if there's any doubt. But, when you talk about two objects meeting in abstract space, you have no way to reduce that further beyond the symbolic meaning of the words. However, the words 'meeting' in terms of abstract objects meeting and 'meeting' in terms of physical objects meeting cannot mean the same thing since by the mere fact that the former has no physical reference like the latter. Doesn't that mean that the former 'meeting' is just an empty word having no real meaning? Couldn't you just say two objects 'snubblebun' together and wouldn't that mean basically the same thing. The word 'snubblebun' has no physical reference, and the term is empty in terms of physical meaning. It seems that's about all you can say about two objects meeting - they are 'snubblebunning' together."?
You see Alan, if someone made this claim, they would be perfectly correct.
H: " Who's mathematical abstract reality? Is everyone's reality the same as everyone else's abstact reality?" Two meet: they are different; but one meeting.......
The objection is that we haven't even defined what an mathematical abstract reality is. Definitions require some reduction in order to be sensical. If we just want to form a belief from some abstract stance, that's fine. There's nothing wrong to holding beliefs in abstract things. But, if we want to precisely define something such that the definition helps us to better grasp what it is we are saying such that there is no confusion, etc, then we need our definition to lock in on some kind of consistent usage. An abstract object is always at least a mental object held by someone's belief system, and if that's what you mean, then you have to state it and explain how consistency can occur from one person to the next. Platonists try to explain mathematical truth by saying that abstract objects exist outside the mind, and that when talking about an abstract mathematical reality, we are indeed talking about a universe of such objects. This is a better definition that lays the groundwork for a mathematical definition of truth since it doesn't rely on human images which are subject to change from person to person, as well as be distorted by the person's mental composure (e.g., drug use, alcohol use, etc).
H: "This might seem unfair to throw these kind of issues in the middle of a simple mathematical attempt to set-up an abstract situation, however this is the result of trying to establish a definition of mathematical truth if one chooses to do so." You appear to have substituted "corespondence with the facts" with "correspondence with rules of math-world".
No. I am concentrating on a Tarskian framework by which to understand how to define mathematical truth. The truth conditions of mathematics is entirely an epistemological issue, and we don't need to discuss those issues since this is what mathematics are doing - and are doing it very well. However, when talking about mathematical truth ontology, then quantifiers are needed in order to specify how the T schema can be a definition of truth, and this is where one runs into trouble. As you can see from 'M2', it quickly moves toward the path toward Platonism, although one could take a nominalistic approach to mathematical truth and say that mathematical abstract reality is a 'particular' in each person's imagination. This is a valid approach, and one in which many philosophers have taken. The definition of mathematical truth for the nominalists will be highly focused on particulars and the universals (e.g., object, plane, lie, etc) all become abstractions of particulars that people encounter in their lives. That's the beautiful thing about the Tarskian approach, it provides the kind of versatility for a nominalist, or a platonist, minimalist, etc to approach the subject matter of truth definition in a slightly different manner. As you can see, it's because of these many differing positions that mathematical truth becomes a philosophical quagmire and the reason why philosophers of mathematics debate it with so much passion.
So he [Hilbert, sic] kept definitions broad enough to allow concept transferability among different more specified contexts? But in so far as he had defined his rules; these became axioms?
These weren't definitions that he provided or kept broad. He simply used words without defining them. This is allowable in mathematics since math is about truth conditions and just as long as you don't start trying to define these terms you don't get into trouble. It's only when people find out how great mathematics is at creating consistent and useful structures that we start saying "hmm... this mathematical stuff is pretty good stuff. I wonder if we can state that truth is mathematical...". Once that philosophy gets going, then the can of worms have been opened, and that's when we need to carefully look at the issues of proper truth definitions and what we mean by references, etc.
H: " Truth in mathematics remains undefined, but the conditions of truth (truth functionals) is of course the heart and soul of mathematics and logic." I do not think so: a minimaly defined rule is still defined. Truth needs no definition by math; truth is what/ who exists. In math as elsewhere; as you judge, so you are judged.
Excuse me for being frank, but Alan you just have other agendas that you are evangelizing that severly taint your view. The fact of the matter is that you aren't giving any credence to any philosophical position other than your logicist views, and that position is not very representative of the philosophical work made in this avenue. I am giving just the tip of the iceberg reply to these issues. Unfortunately, you have so much interest in logicism, that it is apparent to me that you cannot think straight. In fact, your 'religious talk' is so tainting that it makes you appear a little fanatical. I guess since we are talking about truth, a little of that should come out.
I presume by "truth functionals" you mean "structures in keeping with not contradicting the rules upon which they are built"?
Truth functionals are quantified or connected structures with individual true elements in that structure using functional operators (connectives and quantifiers).