As human beings, you and I are equal, based in a principle of equality. — Metaphysician Undercover
Aristotle developed this idea he discovered that matter itself cannot account for any of the properties of an object, and so each individual thing must have a unique form proper to itself. — Metaphysician Undercover
He said the commonly asked question of why there is something rather than nothing cannot be answered, and is therefore a fruitless question. — Metaphysician Undercover
When we’re discussing the ‘=‘ sign we are by definition discussing a symbol which denotes strict identity. — Wayfarer
The form is what something has to take in order to exist. — Wayfarer
Wikipedia says the sign is "used to indicate equality in some well-defined sense... In an equation, the equal sign is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value." — Metaphysician Undercover
Do you recognize that in Aristotelian physics, each individual material object has a particular form which is unique to itself, and this is expressed in the law of identity? — Metaphysician Undercover
Familiarize yourself with the law of identity. It states that "same" identifies one thing and only one thing. There cannot be two things which are the same. Then, take a look at how the = sign is employed in an equation. Clearly the right and left side of an equation cannot both represent the exact same thing, or else the equation would be completely useless. — Metaphysician Undercover
And, I think jgill agreed with me on this point in that other thread as well. — Metaphysician Undercover
So long as you assume a perfect alignment in instances of value structure and possible meaning, then both sides of the equation 4=4 are ‘the same’. — Possibility
The two big objections to Platonism that arise from conversations like this are that Platonic objects lack clear identity conditions and that the ontology is profligate, a crowded slum, what Quine called Plato's Beard. Reducing every object to Math should answer both objections. — Pneumenon
That is exactly what I said. — Wayfarer
When we’re discussing the ‘=‘ sign we are by definition discussing a symbol which denotes strict identity. — Wayfarer
There is something really absurd here. So, you're saying, that in the expression A=A, that this expression only refers to particular instances of 'A'? That in order for 'A' to be 'A' then we have to refer to a particular instance of 'A'? That when we say, 2 + 2 = 4, that you're saying 'hang on! Which individual instances of '2' are you referring to?' — Wayfarer
It's not a matter of 'recognising it', this is something that I have only ever read in your posts. If you provide a reference I'd be obliged. — Wayfarer
From the perspective of appearances of symbols you have a point. Clearly, 2+2=3+1 displays symbols on either side that are not the same as symbols on the other side. So the two sides are not "the same" in this sense. But this is a triviality among mathematicians - and the general public - who associate with each side a mathematical entity, the number 4. Likewise, Four=4 shows different symbols representing the same mathematical item. However, I believe your position exceeds these parameters and is somehow more "fundamental". — jgill
This seems like a silly game of distinction without a difference that could only appeal to intellectual descendants of medieval scholasticism. But I could be wrong. — jgill
I will say that logic, like mathematics, like Shannon information, is not about meaning - meaningfulness is assumed upon use. It’s about the relation between signs (not things) within a specific value system. The equation is ‘possibly meaningful’ only within that system, in which both sides represent the exact same value, regardless of any particular instance, and regardless of its possible meaning. So long as you assume a perfect alignment in instances of value structure and possible meaning, then both sides of the equation 4=4 are ‘the same’. In reality, it’s more like a six-dimensional ratio (0, 0, 0, 0, 4x, 0) = (0, 0, 0, 0, 4x, 0), with only some of the redundancy removed - this equation 4=4 is entirely redundant in logic, mathematics and Shannon information theory. It has meaning only when the sides are NOT identical. — Possibility
The "=" in mathematics signifies that two distinct things have the same value. It does not signify that what is on the right is the same thing as what is on the left, — Metaphysician Undercover
To have an identity is to be identifiable as a unique and particular individual. — Metaphysician Undercover
(my emphasis)Philosophical cognition is the cognition of reason by means of conceptions; mathematical cognition is cognition by means of the construction of conceptions. The construction of a conception is the presentation à priori of the intuition which corresponds to the conception. For this purpose a non-empirical intuition is requisite, which, as an intuition, is an individual object
Analytical judgements (affirmative) are therefore those in which the connection of the predicate with the subject is cogitated through identity; those in which this connection is cogitated without identity, are called synthetical judgements.
We must ensure that the mathematical axioms which we employ conform to reality or else they will lead us astray. Therefore it is actually necessary that we do change mathematical axioms as we try and test them — Metaphysician Undercover
So, again, you're saying that every occurence of 'A' is unique? I still think you're confusing the law of identity, with the meaning of individual identity, which are different subjects even if related. — Wayfarer
I did take the time to read your argument on essence, substance and so on. As you note it is replete with difficulties, ambiguities and aporia. This is a deep problem with Aristotelian metaphysics, generally - the difficulty of arriving at any ultimate definition of the fundamental terminology, I think due to the inherent limitations in reason itself. But, it's still worth studying and I appreciate the time you've taken to spell it out. It's one of the subjects I'm trying to find time to understand better. — Wayfarer
But if we admit an essence that is my father, he loses his individuality since some other person with the same (identical?) properties would also be my father. — Garth
Certain proofs in mathematics hinge on the dissolution of separate identities. For instance, the proofs on this page about lines tangent to a circle presuppose the existence of points with certain accidents. It is through this method that the contradiction necessary for the proof is shown. This reflexively shows that the points themselves cannot have the accidents which were assigned to them and thus the essence of the points of tangency is grasped. The proof equivalently amounts to showing that these points are the same. — Garth
Plato's mistake, it seems, is not noticing that identity only arises insofar as objects are not the same. It is an instrument of abstraction or speculation. Its persistence indicates an indefinite understanding. This implies it is never really present in complete understanding, actuality, truth, etc. Perhaps he was disturbed by the thought that his own philosophy suggested that we do not really have individuality or self-ness. It may have also threatened some of his assumptions about Ethics. — Garth
Kantian intuition therefore must involve this process of construction and dissolution of identity, not as sameness but as arbitrary differences which ultimately prove insubstantial for the concept. — Garth
Kant seems to use Identity to mean sameness, or more specifically that to deduce two things as the same is to show that they share the same identity. This is further supported by Division I, Endnote 1. So even Kant doesn't really distinguish sameness from identity. — Garth
You appear to suggest that mathematical axioms are similar to theory in physics. String theory, however, seems un-testable at present. Does it then lead us astray? If you were to say it does, how could you possibly know? How might you test the Axiom of Choice? — jgill
fishfry refers to math as a game, and it certainly is that. But a practicing mathematician may lose that perspective and math may assume a kind of non-physical solidity and seem "real", even when it's not obvious that it may be related to physical phenomena. Similarly chess probably seems "real" to serious devotees. Incidentally, MU, "pure mathematics" simply means not immediately applicable to the physical world. I've dabbled in this sort of math for decades. — jgill
What's your opinion on my reply to jgill above, Possibility? Do you agree that what the mathematical symbols represent are operations? So when we have an equation, we say that the operation on the right side has the same value as the operation on the left side. And when we say that 4=4, the symbol 4 refers to a grouping of individuals, and we say that one grouping of four has the same mathematical value as another grouping of four. Therefore relative to mathematical value, a grouping of four is "the same" as any other grouping of four, but relative to identity, the two groups are clearly not the same. Notice how I refer to the "grouping" of four, because this is an activity, an operation, carried out by the sentient being which apprehends the four individuals as a group of four. Likewise, to apprehend one thing as an object, an individual unity, is an operation (individuation) carried out by the sentient being which perceives it that way. This fundamental act of individuation is the basic premise for mathematics. Therefore the axioms of mathematics need to be well grounded in the law of identity which stipulates the criteria for being an individual. — Metaphysician Undercover
I don't consider any such human activity as a game. Games are played for entertainment, and in general, the goal is to win. — Metaphysician Undercover
You must use a different definition of "game" — Metaphysician Undercover
Anyway, fishfry goes beyond your definition of "pure mathematics" to claim that "You can, if you like, view the entire enterprise as an exercise in formal symbol manipulation that could be carried out by computer, entirely devoid of meaning. It would not make any difference to the math." — Metaphysician Undercover
It seems like you're not familiar with the law of identity. — Metaphysician Undercover
So, if 'A=A' is meant to represent the law of identity, then A represents an object, and both instance of A represent the exact same object, and '=' signifies "is the same as". A thing is the same as itself. — Metaphysician Undercover
Clearly, "is the same as" does not mean the same thing as "is equal to". — Metaphysician Undercover
The law of identity refers to all things, and I'm not sure what you mean by "individual identity". The law of identity is a fundamental ontological principle which represents the uniqueness of a thing. — Metaphysician Undercover
What I mean by ‘individual identity’ is ‘the identity of individual particulars’. — Wayfarer
But that has no bearing on the symbolic representation 'A=A' because in that case, we're not referring to particular beings, but to symbols. Same with mathematics. Symbols are abstractions, but due to our rational ability, they have bearing on the world. — Wayfarer
I’ve found that the term ‘object’ - denoting a consolidated focus of thought or feeling - is often freely applied to physical objects, events or concepts. I find this ambiguity leads to much confusion, and I’ve had numerous discussions with other contributors to this forum regarding the dimensional distinctions between the relation of self-consciousness to, say, an actual object, an operation/event (eg. grouping), a symbol for the concept that represents the value/significance of an event, and meaning prescribed to that symbol. — Possibility
But mathematics and logic, like computer information systems, are often treated as closed conceptual systems, with any qualitative relations (necessary for the system to be understood) assumed and consolidated: ignored, isolated and excluded. So a ‘mathematical object’ refers to the ‘individual’ symbol for a concept that represents consolidated value/significance of an event - any instance of which is a subjective, temporally-located relation between an observer/measuring device and qualitative relational structures of measurement/observation. But within the isolated conceptual system of mathematics (which effectively assumes and then ignores an alignment of underlying relational structure by abstraction), a ‘mathematical object’ would abide by the law of identity. This from the Wikipedia entry on Law of Identity, referring to violation: — Possibility
The Law of Identity applies only in a logical, abstract (closed) system of thought or language. Any ‘mathematical object’ is interpretable in reality only by a self-conscious observer in a qualitative potential relation to both the symbol (to prescribe qualities of meaning) and the event (to attribute qualities of sense or affect). The moment you relate the Law of Identity to anything outside of logic - ie. once you cannot assume an alignment of sense or meaning in discussion - you risk violation. — Possibility
Sometimes games are played for money or prestige. The professional mathematician finds his activities entertaining, frequently fascinating, and he definitely likes to arrive at a result before others. He likes to win. — jgill
Nope — jgill
I don't know what that means. Your example was triangles. Triangles are identified up to similarity by their angles; and up to congruence by the lengths of their sides; and identified uniquely by their congruence class and position and orientation in space. — fishfry
If people are playing the same game, then they hold the same goal as the object of that game. If all mathematicians do not have the same goal, then they are not playing the same game, and we cannot describe mathematics as "a game" — Metaphysician Undercover
Pure mathematics is more like an art. — Metaphysician Undercover
Here's a quote from Stanford: — Metaphysician Undercover
The point was that an abstraction, a Platonic Form, does not fulfill the conditions of the law of identity — Metaphysician Undercover
Platonic form of the isosceles triangle... — Pneumenon
Consider that when you think about triangularity, as you might when proving a geometrical theorem, it is necessarily perfect triangularity that you are contemplating, not some mere approximation of it. Triangularity as your intellect grasps it is entirely determinate or exact; for example, what you grasp is the notion of a closed plane figure with three perfectly straight sides, rather than that of something which may or may not have straight sides or which may or may not be closed. Of course, your mental image of a triangle might not be exact, but rather indeterminate and fuzzy. But to grasp something with the intellect is not the same as to form a mental image of it. For any mental image of a triangle is necessarily going to be of an isosceles triangle specifically, or of a scalene one, or an equilateral one; but the concept of triangularity that your intellect grasps applies to all triangles alike. Any mental image of a triangle is going to have certain features, such as a particular color, that are no part of the concept of triangularity in general. A mental image is something private and subjective, while the concept of triangularity is objective and grasped by many minds at once.
There are cliques within the broad structure of math in which participants work towards common goals. I was in such a clique. — jgill
Since leaving my clique years ago, this is how I perceive math. I was never a good game player since I enjoyed going off in imaginative directions and doing my own thing. — jgill
We've been discussing the nature of symbolic expressions, such as a=a, with some tangential discussion of the platonic forms. — Wayfarer
The law of identity is not concerned with symbolic expressions, because it stipulates that the identity of a thing is within the thing itself — Metaphysician Undercover
You know that this ('a=a') is just a symbolic representation of the law. So the symbols need to be interpreted. What 'a=a' represents is that for any object, represented as 'a', that object is the same as itself.
Therefore it's not saying that 'a' doesn't have an identity, it's saying that the identity of the object represented by 'a', is the object represented by 'a'. — Metaphysician Undercover
We have clearly been on different pages here. — Metaphysician Undercover
And that's because you're treating the 'law of identity' as an ontological issue concerning the 'essential nature of beings.' — Wayfarer
Yes, but that is a much deeper problem, in some ways. You're talking about ontology, the nature of being. But the debate started with the argument over whether, in the expression a=a, that the 'a' on both sides of the '=' is the same. I'm saying, of course it is, and that the identity of 'a' is fully explained by its definition. I'm not talking about the being or essential nature of a, because 'a' is a symbol. — Wayfarer
The question I was asking at the time was whether numbers (etc) meet 'identity conditions'. And actually your answer was 'yes, but this is not relevant to the 'law of identity'. — Wayfarer
The point remains, however, that in the domain of symbolic logic, maths, and everyday speech, the identify of the symbols used - letters, numbers and so on - is fixed in relation to a domain of discourse. — Wayfarer
I don't think anyone was ever talking about the status of the symbol, 'A'. — Metaphysician Undercover
I think that the law of identity is actually an attempt to produce a closed system of thought. It is a prescriptive rule as to how we ought to use terms. Of course, as soon as a rule is imposed, there will be violations, that's the point of producing the rule, to distinguish violation from non-violation, and attempt to clear things up. But without the law of identity being enforced, there is freedom of ambiguity, and equivocation, as you describe. — Metaphysician Undercover
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