Aleph-null is definitely not a quantity above all others, for instance. — MindForged

Thank you for being rational. As you can now see, thinking infinitity as a quantity whatsoever is against any basic calculus teaching.

How is this different from what I said? — MindForged

Is different in a very significant way. Because CONSIDER something as something else(with reasons to do that) is a thing, DEFINE(as you said) something is another. Citing you:

that's not just the singleton of a set. It was the DEFINITION of the number one in set theory. — MindForged

Numbers are not defined in set theory. They are considered elements of sets. Numbers are defined by induction in Peano Arithmetics, postulating 0 as a number and posing the axiom of succession.

everything is a set in the set theory construction of mathematics - obviously) — SophistiCat

This is naive set theory darling. Thank you for your opinions directly expressed on my communication skills. I am sorry you don't understand set theory, but that's not my fault.

how we measure, by units — Metaphysician Undercover

Yes, and what do you measure my friend? Quantity. Hence there is quantity yet, and you measure it. That fact that the methods and units of measurement may be relative(as you correctly say) does not make this less true, i.e. quantity is presupposed REALITER as what is to be measured.

Which basic postulate of physics states otherwise? — Metaphysician Undercover

Independently existing matter, independently from particular modes of perceiving it, and that actually causes any perception to happen.

"the degree" may be arbitrary. The premise of "the distinguishability of degrees" misleads you. So we commonly divide space by degree — Metaphysician Undercover

The degree as a reference of measure is arbitrary, but clearly the degree of complexity of bacteria is inferior to that of an dog as an organism. No one divide space by degree man, that is nonsensical. Topology does not consider differences of space by degrees at all.

The differences in the things, as you say, is a difference which is not merely spatial, but involve matter, hence degrees. Because if you can not recognize the difference(with a possible measure) things are not at all distinguishable, except by logical or topological characteristics. Unfortunately, we are sensible beings.

If Zero is defined as the empty set — MindForged

This is the condition to CONSIDER the 'singleton'(I trust you on the term) as the number one, but not to DEFINE it. Numbers are defined by a postulate and succession: see Peano Arithmetics( 0 is number(POSTULATE). If n is a number, then n+1 is number.(AXIOM) Then 1 is a number, as 0+1=1.)

I'm still confused about what you meant when you said I had a medieval theologian type of thinking... — MindForged

Not referring to you in particular. By 'medieval thinking of infinity'(and conception more or less related to it) I mean a very precise thing that I say to you.

Thinking infinity has THE QUANTITY OF WHICH NO QUANTITY IS GREATER is the MEDIEVAL conception of infinity.

Since Kant gave (1781!!) an explanation on why this is wrong, I cite from him(Guyer Wood translation, p.472)

«I could also have given a plausibleb proof of the thesis by presuppos­ ing a defective concept of the infinity of a given magnitude, according to the custom of the dogmatists. A magnitude is infinite if none greater than it (i.e., greater than the multiplec of a given unit contained in it) is possible.58 Now no multiplicity is the greatest, because one or more units can always be added to it. Therefore an infinite given magnitude, and hence also an infinite world (regarding either the past series or ex­ tension), is impossible; thus the world is bounded in both respects. I could have carried out my proof in this way: only this concept does not agree with what is usually understood by an infinite whole. It does not represent how great it is, hence this concept is not the concept of a maximum; rather, it thinks only of the relation to an arbitrarily as­ sumed unit, in respect of which it is greater than any number. According as the unit is assumed to be greater or smaller, this infinity would be greater or smaller; yet infinity, since it consists merely in the relation to this given unit, would always remain the same, even though in this way the absolute magnitude of the whole would obviously not be cognized at all, which is not here at issue.»

I will, a time or another, explain in details how mathematical infinity relates to what Kant meant by Transcendental CONCEPT of infinity, i.e. the relation between math-logical and epistemological infinity.

This is the 'singoletto'(I don't know the English word, kind of 'single set') not of a number.

the cardinality and ordinality of sets which can be put into a function with a proper subset of themselves — MindForged

The cardinality and ordinality are not at all variables which can be 'put into a function'. You are very confused.

Finite Numbers can be defined in terms of the cardinality of sets in just the same way. — MindForged

This is not even wrong. It has no sense. Numbers are not defined otherwise than by postulating a number and stating which operations preserves that being a number. See basic, high school Math(Peano Axiomatization of Arithmetics).

Of course they are. — SophistiCat

There is no reply to a deep confusion like yours. BELIEVING numbers be sets is at least a funny thing.

Quality is the degree of excellence of a thing. — Metaphysician Undercover

Frankly, a definition so senseless as all the medieval definitions were. You are deeply confused. You can quantify on something, by selecting somehow a unit of quantity, only if there is yet a quantity independently from that selection.

Furthermore it does not give us the 'capacity of measuring', because, in fact, a measure(as you may intend, for the use of terms by you is ambiguous: physical measure? or which one?) presupposes a mode of measuring, and so presupposes a quantity on which that method is to be applied. In physics, this is matter, or whatever you may call it.

But since you are so darkened on thoughts about what a quality is I will give you a fair dispute, stating better your naive argument and than responding to this peter version(be honored: the argument is by Kant):

«Reality in appearances(i.e. matter as we perceive it) always has a magnitude(a scale of degrees, i.e. quality), which is not, however, encountered in apprehension(you do not apprehend the space in which it is by itself) , as this takes place by means of the mere sensation in an instant and not through successive synthesis of many sensations, and thus does not proceed from the parts to the whole; it therefore has a magnitude, but not an extensive one.» p.290 of Critique of pure reason (guyer wood)

Then he continue:

«Now I call that magnitude which can only be apprehended as a unity, and in which multiplicity can only be represented through approxima­tion to negation = 0, intensive magnitude. Thus every reality in the appearance has intensive magnitude, i.e., a degree. » ibid.

Now, if something is to be selected as a discrete unity, as you say, it presupposes a quantity on which this selection is operated. Or do you think we actually create quantity by itself, against the basic postulate of physics?

Kant is saying that we do not apprehend it, in the sense that the homogeneity of the synthesis is a presupposition, and also it renders possible at all to select a unity: for a unity is a unity of a manifold, and if there were no homogeneity it would not be the unity of that manifold.

It is only deniable by pathologically affected(in the brain) people that homogeneity is a spatial property. And quantity in general if we have to be cautious, thus not saying quantity in general is space(which is a big assumption), we say that PRESUPPOSES ONLY SPACE, as its parts are external to one another: this sufficing to provide an account of quantity, on the postulate(common to every physical theory that has any sense) that space is occupied by something(I,e. matter), and THUS a quantity is possible at all, insofar as we can distinguish it from the space itself and thus it is possible to be measured with a certain referential unity.

Hence as the scale of degrees(quality) relies on a spatial property, than the degrees do rely on that to. But that property alone can not give us any quantity, for, by a quantity, we do not intend a mere externality between parts, but an OCCUPATION of (parts of) space. Hence Space(spatial properties we are able to detect) and matter are presupposed by any concept of quality that has any sense.

But neither Space nor matter presupposes any detectable quality by themselves. Quality, furthermore, presupposes a RELATION between space and something, which renders possible to detect some spatial properties or, as you seem to prefer, to select from that properties units, in respect to which establish a scale of measuring. This something is matter. Matter does not imply quality(degrees) but the distinguishability of degrees implies matter. But matter presupposes space. Then quality presupposes space. Either you identify space with the properties we can distinguish and classify under the kind 'spatial' and name IT quantity, or you do not identify space with those properties and call those properties 'QUANTITY' it is the same for our question: quantity it is presupposed by quality.

Define explicitly a quality without presuming a quantity man, if you can. You will find you can not, and since quantity presupposes a relation, insofar as a quantity is relation between spaces, then quality presupposes relation.

I add: YOU think relations are comparison, but this is a very special case of relations. I'll give you a fair example of a relation which is not a comparison: a function. You set your prejudices as metaphysical truth, and this just because your ignorance of the advancements or acknowledgements of mathematics and logic.

Exactly. Nonetheless this DOES NOT MEAN NOR IMPLY infinity IS a quantity. Is a relation.

I will open a thread about it, because it is UNBELIEVABLE that people still think(like Medieval theologists and the dogmatics) infinity as a quantity.

The only problem here is that "sets" are based in qualities, — Metaphysician Undercover

This is false. Sets are based on RELATIONS(between something, i.e. a set, and its elements. The empty sets have a relation such that no elements belongs to it).

Therefore set theory does not naturally fit our idea of quantity. So set theory provides a set of axioms which modify mathematics in a way so as to be inconsistent with our natural idea of quantity. — Metaphysician Undercover

These are YOURS(wrong) assertions or opinions about what intuitive quantity should be and how this relates to set theory and about set theory.

This is false. There are sets of numbers, but number themselves are not at all sets, just as there are cluster of berries, but no berry is a cluster.

This is either nonsense or splitting hairs in a way that changes nothing. What counts as a "proof" is determined by the axioms and the inference rules — MindForged

Actually there is a HUGE difference between what you are saying and the actual way in which us in logic prove things.

Your ambiguous use of the term 'determining' it seems to refer to CONSISTENCY.

Let me explain you the very big difference between my(correct) statement and yours(wrong).

If you say that the axioms DETERMINE what a proof is, you are saying that they are CONSISTEN, i.e. that for every propositions you can by means of those axioms, DETERMINE IF that proposition is or is not a tautology.

Now, such was the view of the old proof theory purposed by Hilbert, PROVED INCONSISTENT by a man, Kurt Godel.

This lead to another proof theory: _Natural deduction by Gentzen(and Sequence Calculus, also by him).

In this new theory(which disposes of both Natural deduction and Sequence Calculus) the proofs are not at all 'determined' by the axioms, but the axioms(if there are any:in ND there are no axioms) RELY ON THE 'NATURALITY' of the RULES OF INFERENCE. It is so different from the former theory that Gentzen could prove CONSISTENCY of arithmetic(Peano arithmetic) in 1936. This is important also because of other aspects, but I reserve myself to expose those aspects only later or requested.

It seems to me that you have not clear that AXIOMS and RULES OF INFERENCE are not at all the same thing; and also(here):

"Coherency" here is exactly another way of saying "non-contradictory". — MindForged

that you have not clear that logical consequence is a SEMANTIC characterization of a set of valid forms, while logical derivation is a SYNTACTIC characterization of s set of valid forms. These are so far from being the same that a theorem(by Godel: compactness theorem) has to be proved, in order to demonstrate their EQUIVALENCE(which is not identity).

What counts as a proof requires one to adopt some set of rules by which to establish what will count as a proof. — MindForged

But not what is a proof, yet what it will be considered as 'closed' in regards to the set chosen in respect to the operations and relations defined on it.

But the reasoning employed in the metatheory (what we're using to reason about the construction of the logic in question) doesn't have some inherent correctness to them, — MindForged

This is somehow unclear, for Godel's second incompleteness theorem is taken in some context as stating that the consistency of a theory can only be proved in a stronger theory.

one just ends up presuming some set of inference rules and axioms — MindForged

And this is false, for axioms are NOT presupposed at all in actual proof theory. So the conjunction is false.

So for example, classical logic can be constructed from a boolean algebra, as the two are basically equivalent, so we see that a boolean algebra of sets naturally gives us a certain kind of logic (and the reverse can be done as well). But we know numerous metatheories exist independently of the others using other set theories and such, but you never get to some independently proven axioms or something. — MindForged

This is correct. But does not imply:

You have to assume something is just off the table to get going. I'm not saying this is a problem, it' — MindForged

Just as the chicken assume nothing at all to generate an egg. If a chicken could reason, it would try to explain the egg thing trying to distinguish products and processes and then trying to derive from those processes, as independent from the univocity of resulting in a egg, other things, verisimilarly of the same kind somehow.

Sure, I took those words to be mean same thing. I used the word incoherent the idea itself is without meaning because I'm skeptical one could even conceive of how it could even be done. The idea of an independent proof of all axioms makes the mistake of forgetting that what constitutes a proof is determined by some set of axioms and inference rules. The inference rules in proof systems are, after all, taken to be primitive. If they could be proven, we would not take them to be primitive. — MindForged

Just a minor remark: primitive notions establish what is an object, in order to distinguishes operations in their properties, but no operations is a primitive notion at all. The fact that our thought is relational is rather factual, and is somehow obtained as an awareness by reflection and maybe something else as a condition, but it is not primitive in the same sense that the notion of 'set' is primitive.

By the way, I think this may be just a minor query, and we agree on the main points.

Thank you for clarifying your thoughts.

In the first case, infinity is a shorthand for a limiting process (the infinity is hidden in the quantifier 'for all epsilon') — fdrake

Since you said correctly that in the definition of limit the notion of infinity is hidden in the quantifier I think you are not confusing the limit to infinity (infinity indicates the graphical correspondent to the behavior of a function, in so far as its values become arbitrarily large) with infinity of the limit(infinity limit a process) I suggest to not use the misleading terms: 'limiting process' but 'unlimited variables bounded in regards to which the limit is considered'.

Also

Nothing to prevent you from adding 1 to infinity. — tom

Of course tom is right(but unfortunately unsuccessful) in preventing the tedious error of conceiving of infinity as a set of numbers. More explicitly: infinity is a relational concept, and its use as a factor just means the operations operate on variables(and infinite variables[of numbers] plus the unity is of course a not problematic thing). One thing is infinity as relational concept, another is the Symbol of infinity to indicate infinitely many variables.

In the second case - for cardinals - they give the size of infinite sets, so yes they are probably quantities since they represent the magnitude of something. — fdrake

I don't think cardinality offer a quantitative view of Infinity, since it is either a relation between a set and its elements or between its elements and numbers(e.g. a set is D-
infinite iff for every natural number the set has a subset whose cardinality is that natural number) or between sets(e.g. the cardinal of R is bigger than the cardinal of I)

Actually the method of tables rely on the interpretation of the logical connective, in regards to enunciative logic.

Modus ponens is indeed a completely different thing: it is a valid argumentative form. Thus, it presupposes an enunciative theory to be connected to.

No. "Proof" is defined by the axioms and inference rules one adopts. Ergo, there's no way to independently prove the validity of such things because proof and validities are what you get from the above things. — MindForged

There are two errors: 1 proof is not 'defined' either by axioms or inference rules. Proof is the results of applying inference rules by means of axioms. This is not at all a definition, just as the egg is not a definition of the chicken.

2 It is false that you can not prove validity of 'such things'(axioms? rules of inference?) independently(even if your statement is incomplete, because you do not specify independent from what). The proof of a logical theory is obtained by verifying the coherence of axioms, i.e. through the non contradiction principle in a certain form: iff from the set of axioms you can not derive, through rule of inference, a contradiction, then the set of axiom is coherent. You can even proof the independency of some axioms from others by verifying that the same theorem deducible by n axioms+ x axioms is deducible even just through n(or x) axioms.

Of course do not exist an absolutely independent way to proof something, because a demonstration itself is defined as the derivation of something from something which is different, and this IN ACCORD TO A RULE(or how else is to be recognized as a relation of derivation and not a succession?). There is always a reference to establish the recognition of a result from the relative process. And this you say:

So the notion of a purely independent proof, of "laws of thought" or absolute, inescapable presuppositions that need no proof is just an incoherent idea. — MindForged

But not because it is incoherent, but because is IMPOSSIBLE.

I am not saying that rules of inferences are eo ipso processes: yet to us at least there is a correspondent process, through which we infer from something to something different.

The rules do not claim for proof because they are a process. Just as you do not search the proof of the friction causing your car to adhere to the ground, but an explanation of it, which is a completely different thing, and not at all a logical proof.

Being logic the form of thought in general, it is a necessary condition to be SATISFIED in order to have some way at all to distinguish a process and its results. Once satisfied, we have the condition under which proofing has a form at all, recognizable.

Inference are by definition NOT deductions. Ergo they are NOT proofs. They are simply refined conjectures. And that is what Science is all about. That is why Science is NOT Philosophy. — hks

Here is a classic confusion( which MindForged did not make) between a method to obtain a proof and a proof, i.e. between say something THROUGH a language and to say something ABOUT a language.

Inference are what renders possible to distinguish at all a logical difference, and since this is the basis of thought in general, without it would be missing the GROUND to proof something. This is the reason why inferences can not be proved, because by RULES OF INFERENCE a ground to proof is furnished, and THEN inferences are made to obtain proofs, after the logical form has been interpreted, and thus there is an object at all as the material to realize the structure of inferences, and Then to infer.

Inferences are not at all relying on definitions, because they are of a different status(logically): a definition is an EQUIVALENCE(hence it presupposes a logical structure too, and is a PROPOSITION) while INFERENCE is the PROCESS through which from NOT EQUIVALENT propositions, one is obtained from others. And a process is not defined, but IDENTIFIED, such as gravity is not defined ny the gravitational law, but its EFFECTS are recognizable insofar as they are measurable through that law, which is so far to be a definition of gravity that it is gravity that made possible to discover the laws. The laws of course are conformant to a structure of general thought, which is the only postulate we need.

Your deeply confused view is evident in the next sentence

They are simply refined conjectures — hks

in which you clearly confuse an operation on propositions(inference) with a proposition( conjectures are propositions).

Science is so far from being a pretty little thing of conjectures, that conjectures are propositions not rigorously proved, while proofs are propositions rigorously proved and those, in a systematic unity, constitutes what every rational being call 'science'. A bunch of conjectures, without a system in which they may be proved are just grammatical fantasies.

That is why Science is NOT Philosophy. — hks

Being clear that you got wrong giving your account on what science is(or should be in accord to your thoughts of course) I would be curious to hear the difference between science and philosophy, being each science a coherent and consistent system of conclusions obtained through rule of inferences from objective premises. If philosophy is not science is just a fable, as everything with no math and no logic

.

'm not being verbose. You said inferences are by definition not deduction. I'm just asking you what definition are you using, because I pointed an exemple of a deductive argument made by an inference.
But then I realized that an inductive argument contains inferences as well, so an inference cannot be a deduction. You could just answer my question instead of being defensive. — Nicholas Ferreira

This is a very educate behavior and to me an index of a truly inquisitive, instead of rhetoric, nature.

You are totally correct Nicholas: you recognized that our Great Master Hks would like to teach us(or to put blame on our supposed inferiority in regard to his great mind) Logic, but he don't even know the difference between an operation(inference) and the validity of an argument, for both induction and deduction are inferences, and both can be valid, hence they differ logically(i.e. in respect to validity) just in respect to the truth of the premises. He confuses deductive/inductive(which is psychological) and deductively/inductively VALID(which is logical).

41
The claim that
the truth table is the conjunction of the premises implying in the conclusion
— Nicholas Ferreira

does not seem correct. There is nothing inherent in the definition or concept of a truth table that identifies it as being anything other than a tabular representation of the possible binary values assigned to some variable. Consider the simplest truth table:

A | A
_____
T | T
F | F

Which just says the variable "A" has the values assigned to it. There is nothing here about "conjunction" or "implication". Truth tables can further be used to define what certain operators like conjunction and implication mean by showing how the values assigned to variables change when the operator is applied to them. At this point truth tables are used to introduce the notions of conjunction or implication or whatever, but they don't purport to prove anything about these operators. Once you have defined what the operators are, you can construct tautologies that are the functional equivalent of the rules of inference that show that the rules preserve the truth values of the variables they're applied to because they are, well, obviously true as tautologies. The rules of inference then are just short hand ways of constructing these tautologies that are more convenient to work with. — Mentalusion

Here is a person who reflects before writing. Not surprisingly he said something clear, in a very educate manner, and also correct; also with some interesting elements of originality: «truth tables are used to introduce the NOTIONS of...» in a somewhat 'diagrammatic' view of recognition(not merely linked to processes).

Frankly, remembering Hawking, I don't know how to constructively respond to such a stream of consciousness.

infinite sets. Partially defined. No cardinality. — Devans99

it is non sense 'partially undefined'.

But maths tries too do this — Devans99

But not at all. Considering a collection as an object is not considering a collection as finite.

Maths tries to treat these two different object type the same which is an error. T — Devans99

Infinite sets and finite sets are no different object type, for they are both sets.

thats all nonsense IMO. — Devans99

You definitely need to clarify-or to understand- set theory.

Your argument rely on a naive epistemology. In fact your argument, except for the imprecise definition of universe and the wrong definition of time is identical to that stated in the thesis of the first antinomies of pure reason in Kant's Critique(1781 first edition).

Your definition of time is wrong, for 1 time is not a measure 2 is not a series of events, nor the order of a series of event: the order of a series of events is a CONSEQUENCE of time as an effect. Time has been studied, and defined successfully in regards to the direction of events in late 1800 by Carnot, Gibbs, Clausius, Boltzmann and others, by thermodynamical laws.

You need to clarify your thought on set theory, because saying there exists no infinite set is incredibly wrong. And saying that

the set of naturals:

{1, 2, 3, 4, ... }

is partially undefined — Devans99

Is not even wrong: it's non sense.

IE it is not defined so it can never have a cardinality and you can't treat it like a finite set. — Devans99

Finiteness and Infinity of a set are very well defined properties of set in set theory, actually. Your cited sentence it is somehow not grammatical: of course you can't treat natural numbers as a finite set, because it is not a finite set.

Very deep question Amadeus. You found a great issue which many didn't see: the problem of spatiality of sensation. Many just presuppose they are mere qualitative. But this is an assumption presumably false.

If you think sensation can not be spatial, i.e. they are just intensive, then the possibility of anticipate them somehow would be a creation, since they could occur otherwise just because of interaction with real, indipendently existing objects.

By 'intensive' I Mean: they have a qualitative degree in a scale of your perception which is higher than 0, comprehended in that scale, and not above a certain maximum in that scale).

If you think sensation may be spatial, then it is possible, somehow, that by determining space you are determining at least a scale through which sensation would be articulated.

Of course, if this is ever to be, does not mean that you can anticipate something specific, but only a degree of sensation.

Not at all: the possibility of deriving something effectively is what distinguish what we know and what we can not say we know. If your contention were true there would be no criteria to establish in which direction orientates a research with uncertain results, except on some kind of unspecified usefulness.

Furthermore, you're hiding something: if an idea is useful, and if we rest on usefulness alone, there is no other way than casual discovery to search for another, because we may rely on the first occurring useful idea. Using this kind of reasoning, only casualty, and not reasoning, would have been the source of discovery such as calculus(which deepest origin is: how to calculate the area of a circle), its application to physics and the incredible development of technology.

Explicitly you are saying, that(and this is so disputable) we CLASSIFY ideas on the criterion of their usefulness, which, in this case, is COMPARATIVE criterion not a GENERATIVE, as I requested you to give your account on.

You keep changing a lot your mind or at least the way you express your thesis. Maybe is a good sign.

What operation with an indefinite result do you refer to? — Devans99

Any theories you like which produces theorems by inference.

Just because there exists an 'infinite' number of something in our minds, does not imply an 'infinite' number of something is possible. Our minds are simply in error. The concept/relation of actual infinity does not translate to reality. — Devans99

That's false. The first axiom of modal logic (axiom by Alfred Tarski) is: p→◇ p which means: if p is given, than it is possible that p.

Actual infinity (set theory) has not. The first reflects nature, the second does not. — Devans99

That's false. Computer science is based on set theory. Classical mathematics is based on set theory after the development of mathematical logic. And, since you yourself(as anybody who is not insane) admit that classical math brought many results to as, especially in physics, for physics without math is a mythological novel, and since calculus is part of classical maths, it follows that Set theory brought as much as classical maths does, inasmuch this latter is based on the former.

Relations don't exist in the real world, quantities do — Devans99

Quantities rely on a relation i.e. parts external to one another.

That's a potential infinity. Anything related to computers is potential rather than actual. Computers compute over time and have a finite memory capacity so cannot by definition deal with actual infinity. — Devans99

I think you keep confusing the RELATION which infinity is and the RESULTS of an operation, which are not infinite, but indefinite, i.e. as long as you operate you get results, and you make a contrasting view in infinity just on this latter plane.

It is irrelevant whether or not a computation rely on limited faculties, for an abstract method of compute infinitely many proposition there is: compute each single one. The problem is how to DECIDE among those INFINITE proposition those which are tautologies(entscheindigung problem).

Your misleading use of words hides that what you call 'actual' regards variables, but to you that means 'results'; while 'potential' regards algorithms(effective procedures discovered, i.e. something which we HAVE the potential to compute. In principle is antiscientific and antirational to believe there exists some problem which is undecidable not just because we have not yet discover the method to solve it, but because it is absolutely unsolvable. I think you might agree with me on this Rationalism, which opposes itself to this mystical unsolvability.)

While you by potential means 'not real'.

Actual Infinity was introduced into set theory for spiritual not logical reasons. Cantor was very devout and believed God was infinite. He thought the whole trans-finite nonsense was dictated to him by God! — Devans99

Very true, but it is pathologic to deny that the application of transfinite reasoning brought to you ACTUALLY EXISTENT machines, and procured great advances in a large variety of fields in technology.

If Newton had said God suggested him his formulas them would not have been the less(nor the most) true, regardless of the author's believe about them or their source.

In advance, in the original formulation of set theory(with no axioms, hence it was not properly a theory explicitly formulated) you could choose how many elements you wanted, as to consider arbitrarily large sets(infinite elements) and treat them as the object of your enquiry. THIS was the reason that Cantor gave of the actual infinity and not a spiritual one. It is an ONTOLOGICAL one.

Hence the universe is finite. — Devans99

You presupposes universe is the parameters of our relation with it. This is anthrpocentrism and anthropomorphism. Furthermore, saying universe IS a quantity is a very big assumption.

Quantities cannot take on the value of a concept — Devans99

But the concept of quantity is studied in topology. It is not a study ranging the answers to: how big is x? but: which properties define a certain kind of space, if the transformations are continuous? Hence infinity does not mean:

Something that goes on forever — Devans99

But it is a Relational concept as I said many times, and the you, after pointing it out correctly, strangely went far away from it.

There is no hierarchy of infinities. — Devans99

This is factually false. It is just you don't know enough calculus nor enough logic.

The definition of infinity as the larger than anything else precludes more than one infinity. — Devans99

You keep intending infinity as a quantity and not as a relation. Infinity is the REASON why, for some operation, it is true that there will never be a result which would be THE BIGGEST/HIGHEST. It is not that one highest, insofar as unreachable, nor it is this (reificated) impossibility.

you have offered no proof that actual infinity exists. — Devans99

I don't think 'actual infinity' is a syntagma which means more than a medieval use of words, which are offering a distinction(between actual and potential) which has been clarified by the results on computability by Church and Turing. They proved the so called 'actual' infinity to exist, because we can not compute effectively all the tautologies in first order predicative logic, unless it is certifiable if a certain formula is a tautology if it is actually a tautology.

When you will get acquainted with this unavoidable conquest of human thought you may understand how the problems linked to the concept of infinity were very well solved almost a century ago- or at least clearly stated. And you will perhaps modernize the vocabulary.

I might show you a video of some event or another and then press pause and say - what is represented by that frozen frame is a specific instant in time. — ProcastinationTomorrow

This presupposes the vision of other frames, because if you have a vision of just one instant you can not say anything different except spatially.

it does not seem to make sense to view the continuum itself as being made up of such point instants, — ProcastinationTomorrow

This is correct, but not because continuum is not made up of point of instants, but because the Continuum is a logical measure(in set theory: the cardinality of the set of real number) while instants are physical measure(in reference to an arbitrary chosen velocity).

since they do not last for any amount of time, you are not going to create a duration of time by adding them together (0+0=0). — ProcastinationTomorrow

Here's the confusion between logic and physics: instants have duration to us, arbitrarily chosen in respect to some velocity. Moreover there is a confusion between physics and maths: instants are not numbers, and time is not a series of instants, but it is a physical(thermodynamical) effect due to our interaction with the physical world, in respect to the value of what the Boltzmann parameter designates: eggs break but do not unbreak, because our interaction with what Boltzmann parameter designates results in a value big enough to influence our mode of perceiving things(i.e. a mode which distinguishes irreversible events). It is treated as a series, improperly speaking, in calculus, to calculate velocity, and in this case it presupposes space, and moreover PHYSICAL space, since velocity is the distance run divided by the time employed to run it.

Could there be a way of taking an instant in time not as a thing itself, but rather as a way of thinking of a duration of time? — ProcastinationTomorrow

As I said, Time is not duration(a property of things) but an effect(a mode of perceiving things. Duration presupposes identity, because to establish that something is the same thing, but in a successive instant in time, you need logic, and a subjective criterion(structurally objective) of determining an object. This presupposes a referential structure, in which a subject is distinguished by the object which the former is related to and in such a way the the perceiving maintain this distinction recognizable.

Citing Kant's clear, unbelievably ignored, account on this topic

«Whatever the content of our cognition may be, and however it may be related to the object, the general though to be sure only negative condition of all of our judgments whatsoever is that they do not contradict themselves; otherwise these judgments in themselves (even without regard to the object are nothing. But even if there is no contradiction within our judgment, it can nevertheless combine concepts in a way not entailed by the object, or even without any ground being given to us ei- ther a priori or a posteriori that would justify such a judgment, and thus, for all that a judgment may be free of any internal contradiction, it can still be either false or groundless.
Now the proposition that no predicate pertains to a thing that con- tradicts it is called the principled of contradiction, and is a general though merely negative criterion of all truth, but on that account it also belongs merely to logic, since it holds of cognitions merely as cognitions in general, without regard to their content, and says that contra­ diction entirely annihilates and cancels them.
But one can also make a positive use ofit, i.e., not merely to ban false­ hood and error (insofar as it rests on contradiction), but also to cognize truth. For, if the judgment is analytic, whether it be negative or affir­ mative, its truth must always be able to be cognized sufficiently in ac­ cordance with the principle of contradiction. For the contrary of that which as a concept already lies and is thought in the cognition of the objecta is always correctly denied, while the concept itself must neces-
B191 sarilybeaffirmedofit,sinceitsoppositewouldcontradicttheobject.b Hence we must also allow the principle of contradiction to count as the universal and completely sufficient principle' of all analytic cognition; but its authority and usefulness does not extend beyond this, as a sufficient criterion of truth. For that no cognition can be opposed to it without annihilating itself certainly makes this principled into a con- AI52 ditiosinequanon,butnotintoadetermininggroundofthetruthofour cognition. Since we now really have to do only with the synthetic part of our cognition, we will, to be sure, always be careful not to act con­trary to this inviolable principle, but we cannot expect any advice from it in regard to the truth of this sort of cognition.

There is, however, still one formula of this famous principle, al­though denuded of all content and merely formal, which contains a syn­ thesis that is incautiously and entirely unnecessarily mixed into it. This is: "It is impossible for something to be and not to be at the same time." In addition to the fact that apodictic certainty is superfluously appended to this (by means of the word "impossible"), which must yet be understood from the proposition itself, the proposition is affected by the condition of time, and as it were says: "A thing = A, which is some­ thing = B, cannot at the same time be non-B, although it can easily be both (B as well as non-B) in succession." E.g., a person who is young cannot be old at the same time, but one and the same person can very well be young at one time and not young, i.e., old, at another. Now the principle of contradiction, as a merely logical principle, must not limit its claims to temporal relations.' Hence such a formula is entirely con­ trary to its aim. The misunderstanding results merely from our first ab­stracting a predicate of a thing from its concept and subsequently connecting its opposite with this predicate, which never yields a con­tradiction with the subject, but only with the predicate that is combined with it synthetically, and indeed only when both the first and the second predicate are affirmed at the same time. If I say "A person who is unlearned is not learned," the condition at the same time must hold; for one who is unlearned at one time can very well be learned at another time. But if I say that "No unlearned person is learned," then the proposition is analytic, since the mark (of unlearnedness) is now com­ prised in the concept of the subject, and then the negative proposition follows immediately from the principle of contradiction, without the condition at the same time having to be added. This is also then the cause why I have above so altered the formula of it that the nature of an analytic proposition is thereby clearly expressed.» Critique of Pure Reason(Guyer Wood) B190-193

NUMBERS ARE NOT SETS

Numbers reflect reality and they do not include infinity. — Devans99

That is a condition whose first member is a very big, but rather vague, assumption. The second member of the conjunction, i.e. numbers do not include infinity, is misleading, for we consider sets of numbers, but numbers are not sets. Hence the relation of inclusion can not be applied to them. It just does not make any sense.

REALITY IS NOT A SET OF NUMBERS

From: numbers reflect reality and numbers exclude infinity you cannot conclude that reality excludes infinity unless you make numbers and reality equal. The argument is however ill founded because of the above reasons.

WHAT IS INFINITY ABOUT

It's meant to represent physical quantities — Devans99

This is hard to believe, for infinity is a RELATIONAL concept, between an operation and a unity in regards to the possibility of generating, for any given result, another who is greater. Infinity IS NOT QUANTITATIVE.

Moreover, this wouldn't follow anyway:

so it should be physically constructible — Devans99

Because we use theoretical entities to represent physical ones, but this does not mean that we physically construct theoretical entities.

Furthermore, constructibility is not possibility, and this is so counter the use of the term that actually what is constructible presupposes the concept of possibility, but not the other way: the concept of a world in which the laws of physics are inverted is possible, for it is not self contradictory; but it is not constructible insofar as we have yet understand its possibility(logical possibility). You need to distinguish in greater detail, because it is never clear one thing, that I am going to ask you explicitly:

WHAT DO YOU MEAN BY 'INFINITY'? Since I gave a very clear and unambigous account on my view, and you, so far, did not.

It would be pure magic if actual infinity exists so that's why it's not found in nature. — Devans99

Again: existence is either a logical quantifier or an attribute of reality. If it is a logical quantifiers it does not distinguish differences in object, and, in principle, the possibly bondable variables are infinite. It's just misleading and nonsensical distinguish actual and potential infinity. It makes sense distinguishing logical, mathematical infinity and physical infinity, which we CAN NOT ASSERT NOR DENY, BECAUSE OF OUR WAY OF KNOWING THING, since it is conditioned de facto, hence the relation between our knowing subject and its objects is one of restriction and not one similar to that of infinity.

Claiming to be magic the «existence of actual infinity» it's just rethoric, without distinction of what do you mean by existence(logical or ontological) and between logical,mathematical and physical infinity, instead of actual and potential(a concept rather vague, which in fact you carefully decided to never expose).

What about all the paradoxes of infinity? Hilbert's Hotel for example. Utter madness. You can't really claim such a hotel could exist? — Devans99

Again: first: the so called hotel is a rather confusing example to explain CANTOR'S Hierarchy of infinities, as intended in set theory as cardinality of sets. It is a theoretical account of problems like: even numbers are infinite. odd numbers are infinite. Hence integers are double infinite. And it is a successful account of explain it. Integers have a. cardinality of Alef, and whatever set of number which may be corresponded 1-1(biunivocally) with the set of integers is infinite of a value Alef, i.e. has cardinality equals to that of the set of integers. Other sets of number, e.g. real numbers, has a cardinality bigger than alef. Hence the idea of a hierarchy of infinities. Read something of set theory and Cantor. It is unbelievable that you don't know anything about set theory, which is the accurate theoretical set of infinity, more than 100 years after Cantor's pioneering work.

You are the one with the irrational belief here. Infinity is magic. Burden of prove that it exists is on you — Devans99

This again is mere rhetoric. I offered a distinction and an exposition of the treatment of infinity in set theory. You keep going on because you didn't clarify your use of terms, and on that I have little burden, except inviting you to consult the actual RESULTS obtained in the treatment of infinity, which you, ignoring, are keeping to treat poetically and mystically, claiming that someone say that it(what, a concept? say that a CONCEPT exists is a grammatical, not philosophical, error) exists is just a straw man fallacy.

As we have already demonstrated, new things cannot be deduced, because things that can be deduced from facts we already have, are not new, they are derived. But the thought that resulted from that "inspiration" (or vice versa, I'm not sure :smile:) is an original one. — Pattern-chaser

What about the identification of new axioms of infinity in mathematical logic? Do you consider them new or just derived from the preceding ones? And what about theorems?

I like the «inspiration arises from thought» scenario.

Strangely, he then went on to suggest that the notion of a 'missing colour blue' is an idea that is not just a connection between existing ideas. Nobody can work out why he did that, and personally I don't agree that it is a new idea — andrewk

Hume does not say that would be a NEW idea, but that it would not correspond to an effective sensation. Thus, being an exception to his established general law that: every idea has a correspondent sensation.

The funny thing here is, that much later(calculus, set theory; then labbra calculus) it emerges that the concept of a function as correspondence presupposes concepts(specific domain and codomain).

MATHEMATICAL, PHYSICAL, LOGICAL INFINITY

Actual infinity, if it existed, would be a quantity greater than all other quantities, but:

There is no quantity X such that X > all other quantities because X +1 > X

The non-existence of actual infinity implies negative actual infinity does not exist. Negative actual infinity has the same structure as past eternal (in time):

{ …, 2015, 2016, 2017, 2018 }
{ …, -4, -3, -2, -1 } — Devans99

∞ + 1 = ∞ implies
1 = 0 — Devans99

This are the two argument you set forth. I explain why they are incorrect.

Mathematical infinity is RELATIONAL: infinity is a property of the relation between an operation and the unity selected to operate on. Thus infinity OF THE OPERATION is non sensical, because there is no quantity of an operation except you identify operation with its results. In this case, there would be INDEFINITENESS of replying the operation or applying others, but no infinity. Infinity regards the results: there is none of the results that is bigger than ALL the possible results.

Your first argument is incorrect, because it treats infinity as a NUMBER.

Your second argument is incorrect because of the above reason and because It mixes mathematical infinity with others conception of it, concluding form the impossibility of an actual infinity in maths and from the impossibility of a negative infinity in Physics, the Logical impossibility of an actual infinity(which is factually false after the works of Church, Turing, Godel among others).

ACTUAL INFINITY would be LOGICAL or PHYSICAL. In so far logic concerns variables, being their infinite(see the results of Church Turing) there is an actual infinite in logic, and this is so true that the first order predicative logic is undecidable. In so far physics is concerned, given we do not know completely the universe, and that is because we need to refer to our conditions as observers(it doesn't change the argument if you add instruments to observe), being those conditions relative to our subjective constitution, an actual infinity does not make sense. Yet, we can't even say the universe IS finite, because the boundary under which we investigate it are relative. Hence, we could say: universe is not bounded by itself to the same conditions we are bounded to in order to know something about it, while interacting within it.

SUMMARY

I think I answer 1: why your arguments were wrong;

2: it is not to be constructed as an object, but it is a rule to follow in order to not stop constructing: here is INDEFINITE, not INFINITE. Furthermore it cannot be constructed because it is a RELATION: you can construct terms of a relations, following the conditions stated by the relation only AFTER recognizing the relation as a possible way to relate terms.

3 Right: physical actual infinity is not claimable to exist. But this imply only that physical world is not infinite(in so far as we can understand natural phenomena) , not that it is finite(being infinite or not, we interact significantly with a certain kind of phenomena: hence it may be that there is actual infinity in nature, but the question itself is nonsensical to us, because we consider nature as it has effects on us, not in regards to how big it is).

4 It's just the opposite: it is perfectly logical(cfr. Leibniz, and recently Church, Turing, Godel et alii), and only because logic is insufficient to account for our view of reality It is not correct to infer from logical infinity a physical infinity.

5 Existence is either a logical quantifiers with reference to variables(which are infinite but we cannot say they exists: they are just logical variables that stands in waiting for an interpretation) or a physical existence, and of course not a single existence it is perceived somewhat infinite, nor we have a complete account of everything that exists(for a classical writing on the topic see Quine, From a Logical point of view, essay 1: "What there is?")

GREETINGS

First, I want to thank Devan for this and eventually others comments, which are always guided by a true inquisitive interest and great respect throughout the conversational form.

Now I answer.

PLATONISM

Citing yourself from the post you linked:

«I believe you are correct; our minds seem to link existing concepts and map concepts across domains rather than creating new concepts.»

We share the same way of thinking here, even if I'd prefer to call the independently existing: schemes rather than concepts, as concepts presupposes an instance of non contradictions which is not required in a more general theory of inference, e.g. lambda calculus. By schemes I intend somehow an instruction to refer, within a general structure of which each cognitive type(e.g. man) would be a realization, the recognition of a subjective unity and an objective one to different patterns, which differs in kind because the differences between those unities need to be preserved.

SOMETHING NEW 'ABOVE' THE SUN

I didn't get if you f r o m this infer the impossibility of the r e c o g n I z a n c e of new concepts, which thing I think is possible just because the difference between our intuition(which is sensible, and need some physical condition to operate effectively) and an intellectual intuition(an intuition which would p r o d u c e directly what it would apprehend).

I cite you again from the thread you linked me to:

«Zero came from consideration of emptiness. Infinity from consideration of the very large.

Can anyone refute this with an example of a genuine new idea?»

Zero, in Peano arithmetics, is just postulated. It does not come from consideration of emptiness, but from the consideration of a neutral element in the operation of succession. The postulate states that 0 is a number. Then there is an axiom, which states: if n is a number, n+x is a number. This axiom closes the set of numbers in respect to the operation of succession. Hence, 0 is a new idea. I argue that every postulate is a new idea. Also hypothesis are new ideas: infinity does not come from the consideration of very large, but from the consideration of the unconditioned nature of the relation between an operation and the unity used to obtain results.

I explain better. If I add one grain of salt to a n quantity of grains of salt, I obtain n+1 grains of salt. But the Operation of addition IS NOT conditioned by this result, neither by the unity(here: a grain) nor by which represents the unity as a scale of measuring the resulting quantity(here: the grain OF SALT). Hence the concept of infinity.

HOW MEMORY IS DISTINGUISHABLE FROM SENSES?

I would have said 'senses or memory'. All thoughts trace a heritage back to senses or memory. — Devans99

Can you please explain this further? I think we say we remember something(potentially) only if a sensation reached a certain degree both of intensity and of complexity. If I get hurt by something which I didn't see before nor after the hit, I would say the remembering equals the sensation, if no further details are added: e.g. I am in a bar, from that point I will remember THE BAR as the place were I was hit.

TIME AT ISSUE

I'm afraid I do not agree with you concept of time (see the other thread). — Devans99

Like I said, I didn't understand what you actually intend with the word 'Time'. In brief, I didn't think is a concept, but a mode of perceiving events depending on our interaction with the physical world. Of course this interaction follows more general law of nature(i.e. valid for beings with a constitution different from that we have) but time itself depend on the subjective constitution to me, and also its eventual development as mode of perceiving things depends on the development of the subjective constitution to which it cohere.

SUMING UP CONCLUSIONS

Hence, I think it is correct to say that OUR KNOWING PROCESS begins with temporal conditions, but not with as time is a mode of perceiving things, yet with time as it would be the manifestation of a physical law of increasing complexity(or, more generally, of an orientation of a sufficiently comprehensive physical process). Following this law, nature will generate some physical system capable of cognitive activity, or, more generally, nature will give the conditions of possibility of the configuration, in which the recognition of the law itself would be possible.

Still, I'm not suggesting that KNOWLEDGE ITSELF, or better: the structures which we recognize as necessary to know, depend on Time: neither on the 'complexity time',i.e. the time which is index of a general law of increasing complexity in the development of life, nor on the 'conditional time', i.e. that we use to determinate a coordinative system to our perceptions.

I think the independency of knowledge from any particular cognitive being has a correspective independency in regards to any particular condition of being known, as it renders possible at all knowledge as a dynamical, regulated process.

First you argue something like:

"All obey to speed of light"

"Speed depends on time"

"Everything depends on time as it obeys to the speed of light".


The speed of light is ARBITRARILY defined infinite, but this is an improper use of the term: the velocity of light is a reference, in so far as it defines a referential PHYSICAL criteria to determine whether something is or is not moving in certain trajectories.

I pass on your silence on your incorrect exposition of the formula of speed.

So infinity is not a mathematical concept. — Devans99

It Is! And you too stated in a comment above, saying that a series is infinte because no terms of it is bigger than every terms of the series(i.e. always exists a bigger one)! I think you have no clear idea of what time is, because sometimes you say it is a series, sometimes you rely on the formula of speed to argue the necessity of the obedience(?) to time and thus(?) a beginning. Time, as perceive, is an effect defined thermodynamically(high school physics).

When you acknowledge Actual Infinity is impossible, the start of time follows logically. — Devans99

Actual infinity is not the assertion that you recognize an infinte number of beings, but the sole consideration of the fact that, being us bounded to certain conditions, being those conditions logically irrelevant, it follows LOGICALLY that an actual infinity it is, under the assumption the the finiteness we perceive is due to unessential condition. Still we can not ASSERT actual infinity, because we need more than logic to say that something exists: logic abstracts from differences in objects, while is because of objective differences that we can talk of knowing something at all.

Speed is distance divided by the time IMPLIED TO RUN THAT DISTANCE. This is not the Time you were talking about(a series) and not only this consideration of time presupposes matter(the difference of velocity is a difference in physical states) but also SPACE: you are unwillingly saying that space is more fundamental than time!

Time is a series: Now (t=0) only exists because t=-1 had existence. t=-1 only exists because t=-2 had existence. So all moments must have a moment prior to them. The only topology that fits is a closed loop IE circular time. — Devans99

You are adding a causal relationship that there is not in mathematical series, and that is not a mere order: it implies a difference in physical state( not mere numerica diversitas).

Infinity is not a quantity: infinity is a concept: as the infinity of skirts(if this be actual) would not be a skirt. In Logic infinity is defined presupposing a concept(i.e. of a set) as the cardinality of this set, in order to establish a hierarchal order on numerical sets(I.e. sets which contains numbers of limited properties). The ordinal infinity, instead, differentiates between finite and infinte sets, being infinity a property of a set, inasmuch it contains a number of elements such as no one is the bigger in regards to the operation which close that set(as you correctly indicate). You atre talking just of ORDINAL infinity.

In analysis this logical acquisition leads to use a hierarchy of infinities to solve calculus problems.

In topology, which does not presupposes time at all, infinity is either some kind of iteration with preservation of a certain kind of space(but time is not the iteration of the operation of succession, as you rightly said) or it is indetermination: you can proceed indefinite in a given space of any kind.

I do not think is necessary to deny negative actual infinity to deny actual infinity(if this be the case), because the regress is stopped because of PHYSICAL reason: we do not not the whole universe, hence our hypothesis is limited, either because of that or because our-certainly- incomplete knowledge of the physical world. While the actual infinity is not concerned with time: actual infinity would not be a series, but simultaneity of existence. But since we can not state anything but relative in regards to simultaneity, then an actual infinity is just the hypothesis that there are infinite many systems, with their respective conditions, operating harmoniously(some kind of Leibnizian theory). One thing is certain: being time and space just condition to perceive things relatively to our subjective constitution, it can not be said that they-or other forms of them- enclose the world. But this just imply the the world is not finite, not that is infinite. Newton considered finiteness or not in regards to matter and denied the infinity of the universe because, if that be the case, we would have not experience the world as we do. But, in regards to concepts, at the question whether the universe is finite or infinte we could just answer: it is neither finite nor infinite, but non finite. To establish the infinity of the physical world we would need to have its quantity determined as a whole, and as a whole that quantity would have to be an object for us. Which is impossible.


Like I said before, The Big Bang is an Hypothesis to explain the actual condition of the universe in so far as we know physics, relatively to our possibility of interacting meaningfully(cognitively if you want) with the physical world, as, presumably, certain physical processes rendered possible our (relative, sensible) subject of cognition. Thus, the Big Crunch it is(but no everyone agrees) a consequence of an hypothesis. While the Big Bang accounts for the actual condition of the universe, with the above mentioned limitations, the Big crunch is more a prevision based on the possible continuity of physical processes, as regulated by the laws we have discovered so far.

I think this is implicitly a definition of beginning similar to that of Kant in the Dialectic: «Beginning of a thing is the recognized difference between a time in which a thing does not exist and a time in which it exists.»

Yet, Kant is more precise elsewhere: this definition leaves out the cases in which we perceive the S a m e thing as changed( to perceive such a thing we need a structure of identification which 1 presupposes the permanent 2 it must be referential, because we perceive change from a certain point of view, which is given).

Well, I would say: if you think beginning as a l o c a t i o n in a certain instant, and not of a recognized difference in reference to an identified state, then there is no reason to think there is a beginning of the world as a whole, because the whole is a mere concept, through which we think a complete regress.

But it is wrong to say: « Time is a series» for a mathematical series is n o t characterized by an o r i e n t a t i o n: you can go back and forth just the same. We do not, as a fact, perceive time this way: we see irreversible event(because of the effect called time, i.e. interaction with is designed by the Boltzmann parameter), t h e n we determine this perception causally, i.e. we consider it as an effect, and a cause is to found. Otherwise, we presuppose a referential structure, i.e. recognition of meaning as Kemp Smith call it, to have the possibility itself to distinguish, in time, what is changed but still was there in other form, and what is present in a certain instant and was not before. With time alone you can n o t establish if you see the same sun every time you look at it, nor you can say there is at all if you are not perceiving it. Hence, time presupposes, as to be meaningful, and not a brutal sequence of events unconnected, a structure(an a priori, as Kant called it).

You are too smart to think that there is an absolute beginning in time, since 1 there Is no difference between mere instant and then no reason to distinguish a beginning instant from others and 2 we perceive time just because of certain constitution of ourselves: time is the effect of our interaction with the physical world insofar as we were capable of describing it in terms of thermodynamic: just because the parameter indexed by the Boltzmann Constant is big enough we perceive an irreversible direction of events. This means: time is relative as it implies a certain perceiving of things. If you by 'Time' mean a law of increasing complexity a n d some kind of correspondence between certain degrees of complexity(however it is to be measured is a interesting and hard problem) and the p o s s I b I l I t y of certain configuration as forms of life, welcome on the ship. If you do I advise you to read Reichenbach, the first to present a clear account of time as an effect(causal theory of time).

The fact that there is an hypothesis, i.e. Big Bang, to explain the actual development of the universe, does not mean that the universe itself begun at that point, but only that just so far our knowledge has been capable of establish an explicative hypothesis.

It is a very interesting question. I suggest to you a lecture: the first Antinomy(abut the beginning of the world or not) in the Critique of Pure Reason you may find pdf online in an accurate edition.

LOGIC AND PHYSICS

The reason why the infinite regress is rejected, in regard to the beginning of physical things, is not logical, but physical: to us the cognitive process involves a « discretion» of matter in unities which can be worked up until reflection let us have awareness of the distinction between ourselves and what is and object to us. This requires some kind of continuity in the physical process of working up data.

Now, if you think the data are selected THEN recognized, you recognize a structure of recognition as basing the process of selection. If you think the other way, you think that recognition is sufficient to elaborate data, but the elaboration is necessary to recognition and independent from it.

In the first case you can't say that the physical WORLD(as a whole) as a beginning whatsoever, because you must first account for the origin of the structure which made to you possible to distinguish two states at all, where a beginning would be the recognition of a thing in a certain state, within the recognition of the absence of any thing like that in a precedent state.

In the second case you can't say that something has or not a beginning, but just that something there is or not, or that something comes before or after(iff you recognize the indipendency of the data, though not existent, in the same form you attribute to them, independently); and this because you have no recognition at all until a certain thing make you recognize something at all. In this case the most accurate thing to say may be: about those data, that I recognized, I can establish a scale to measure them(let's call it: the complexity of them). Furthermore, assuming you know the somehow independent being of them, I can find which complexity is necessary to generate a being like myself(and finding such a correlation alone would be an heroic work).

REFERENTIAL TIME AND SPACE?

Time is, indeed, a definition: it defines the interaction between us and the world insofar we distinguish the property of irreversibility(the infamous example of the egg which breaks but not unbreaks). This is due to the fact that the parameter indexed by the Boltzmann constant designates something the physical effect is enough big to us, that it results in a certain mode of perception of events. Thus time isn't properly a reference, being a reference that indicates any beginning, through the reference of which distinguish a before and an after.

It remains space. Space, in general, presupposes just homogeneity. But, as we perceive, we presupposes other properties, which may varies within our evolution. At now, we perceive the (macro)world as euclidean, whatever the reason may be. There is, however, a book titled "Twelve examples of illusion" (I liked it) in which there is a reference to the work of a mathematician, which disposed a set of exercises to learn to perceive the world in more than 3d..

Hence, not even space has that referential structure that we were searching for: homogeneity alone it is not sufficient to distinguish any beginning(different parts of space are in a relation of difference(or identity) but not of succession).

DEFINITIONS

As Poincaré said somewhere: that the light travels in straight lines is a definition of straight line. That is to say: definitions presupposes a criteria of identifying objects. But in order to do so, we perceive them in space and time. But space and time are not referential, i.e. those alone do not let to identify more than a ordered multiplicity, and not an object for a subject. Therefore, the definitions too rely on a referential structure which does not identify with the criteria of perceiving object, and even less with the actual modes in which they are perceived.

INFINITY

It is a well known topic in mathematical logic and in analysis. Let's just say: Infinity it is not a number, it is a relation: infinity means that no matter how many units you add(how many times you apply an operation): you will never obtain a result, such that the operation can not recur on it. Of course, if the operation is not recursive there is no such problems of establish an end or not to its application through a series. Infinity is, in this sense, a relation between a unit and a correlative operation.

In analysis it is a concept used to establish a hierarchy in respect to the set of natural numbers. I hope nobody thinks a hierarchy has a beginning, just as the laws that regulate the behavior of waves has none.

HAVING NOT A BEGINNING

As in the last analogy, with all its limits, I tend to consider beginning as relative, insofar as it depend on a referential structure which imply a certain mode of recognition. That is: x has begun equals I recognized x and y in respect to z.

It may be the case, that having a scale to measure the complexity of matter, we could establish a necessary(and perhaps sufficient in some respect) value of complexity to the possibility of such beings to become a form of life.