Infinity and Zero: do they exist?
Abstractions do not exist (not in the real world), they are mental devices that we create via The Way of Abstraction. They reflect types of objects that share some common properties. For example, the abstract object triangle does not actually exist (not in the real world) but triangular objects exist - they are instantiations of the abstraction.
In mathematics and logic, we use the existential quantifier(?) in a way inconsistent with the above, often applying it to abstractions. But that doesn't confer real world existence to the abstraction.
Numbers are abstractions, some of which are instantiated in the real world. 4 is instantiated as a property of any group of four objects (properties exist, but only as constituents of the objects in which they are instantiated). ? Is not instantiated; it is an irrational number - a ratio of circumference to diameter - a second level property, not directly instantiated - it is a matematical relation between abstract objects.. It's a useful abstraction but it's still just an abstraction.
Zero is another useful abstraction. 0 has mathematical relations to other numbers, but it is not instantiated in the real world. It refers to an absence of existence, not to an existent.
Now we get to infinity. The arguments I've seen for its existence (e.g. referring to Cantor) only show that it is a logically coherent, mathematically useful, abstraction (like ?, or 0). This does not establish its existence in the real world - an instantiation.
While this doesn't prove the nonexistence of infinity, it does undercut proofs that infinity does exist. At best (for infinity-o-philes) , we should reserve judgment. But I submit it's reasonable to go further, and reject its existence just as we should reject the (actual, real-world) existence of ? and 0, for lack of good reasons to believe otherwise.
In mathematics and logic, we use the existential quantifier(?) in a way inconsistent with the above, often applying it to abstractions. But that doesn't confer real world existence to the abstraction.
Numbers are abstractions, some of which are instantiated in the real world. 4 is instantiated as a property of any group of four objects (properties exist, but only as constituents of the objects in which they are instantiated). ? Is not instantiated; it is an irrational number - a ratio of circumference to diameter - a second level property, not directly instantiated - it is a matematical relation between abstract objects.. It's a useful abstraction but it's still just an abstraction.
Zero is another useful abstraction. 0 has mathematical relations to other numbers, but it is not instantiated in the real world. It refers to an absence of existence, not to an existent.
Now we get to infinity. The arguments I've seen for its existence (e.g. referring to Cantor) only show that it is a logically coherent, mathematically useful, abstraction (like ?, or 0). This does not establish its existence in the real world - an instantiation.
While this doesn't prove the nonexistence of infinity, it does undercut proofs that infinity does exist. At best (for infinity-o-philes) , we should reserve judgment. But I submit it's reasonable to go further, and reject its existence just as we should reject the (actual, real-world) existence of ? and 0, for lack of good reasons to believe otherwise.
Comments (123)
Great question R !!
In a word, I want to say yes. They exist as metaphysical abstracts.
Common analogies (that underlie/describe the physical world), which you probably already know, include:
1. Laws of gravity in physics
2. Engineering/Design formulas for; compressive forces, tensile strength, torsional forces, etc.
3. Musical notation
Also, I agree, I think Omega is useful, yet unknowable.
My question would be how would those metaphysical abstract's relate to the phenomenal world of consciousness and Being, if at all(?). In other words, how useful are they to us, in that context of consciousness and Being. In a paradoxical way, numbers and math can describe the physical world's underlying properties, yet they are purely abstract. I wonder if there is a connection to something...
In my view:
Laws of physics are relations between types of things. Things can relate to one another in ways that can be described mathematically. That doesn't entail independent existence apart from the things that relate in that way.
Same thing essentially applies to engineering formulae- they still are due to natural law.
Musical notation isn't an abstraction, it's a semantics that maps to various aspects of sound.
You refer "metaphysical abstracts" - suggesting abstractions actually exist as (what?) Platonic entities? Why think they exist in this way rather than merely as a mental entity, formulated via the Way of Abstraction?
I'm thinking that it would entail an independent existence. If a given abstract does not exist for the sole purposes of the creation of a particular concrete thing, by definition, it would then be something independent of the thing itself. In other words, if it can be created and/or exists without such abstract knowledge, then what is its purpose for existing?
To that end, using the laws of gravity once more, they do not confer any biological survival value when one can simply dodge falling objects.
Quoting Relativist
I'm not sure I'm with you on that one. Much like written mathematical formula, musical notation represents otherwise-produced symbols, including notation for durations of sound as well as the absence of sound such as rests. I don't see the difference, or even how semantics would play a role.
As far as Platonic abstracts, in my view, the link to timeless truth's is about the closest metaphor we can relate to (in order to be useful), when comparing universal's such as mathematical abstracts; I think they can be non-physical phenomena, yet dependent on consciousness for their apprehension. I'm not a Platonist, but maybe Wayfarer could elucidate there, since Platonism would include the idea that both non-physical and non-mental abstracts exist.
But back to abstracts music and math. When looking at a pre-engineered beam, I know what exists abstractly is a mathematical formula that underlies its physical reality. Conversely, when I read abstract musical notation, I hear music. Abstracts can work both ways.
In set theory (since you mentioned Cantor, who provided the main pre-formal concepts) there is not an entity called 'infinity' (distinct from a other notions such as points on a real extended line or figures of speech such as "as x goes to infinity"). Rather, there is the adjective 'is infinite', and an axiom that entails (with other axioms) certain theorems including the existence of infinite sets.
Existential quantification is not inconsistent with the claim that abstractions are not material objects.
Of course, we can hold that there do not exist infinite sets. But then providing a formal axiomatization for the mathematics for the sciences gets a lot more complicated.
That sounds like Platonism. My problem with ontologies that include platonic objects is that they seem unnecessary. Why posit an independent existence for triangles, when triangles can be accounted for as constituents of triangular objects? Further, how do triangles exist independently? How do they get connected to objects? Can the connection be severed? This makes it even more unnecessarily complex? Can they replaced with squares simply by replacing the connection?
Quoting 3017amen
The notation is interpreted by a musician, analogously to a reader interpreting print words. Words refer to objects, concepts, actions etc, while musical notations refer to the various aspects of sounds you mention. The sounds can be reproduced on an instrument, or merely interpreted within the musician's mind.
Quoting 3017amenBoth ways are consistent with the way of abstraction. We mentally consider a set of attributes common to all triangles to form the abstraction in our minds, then reverse the process, adding back concrete elements.
Sets are abstractions. Creating abstraction just means conceptualizing. My point is that abstractions don't actually exist except as mental entities. Mathematical abstractions are useful because they entail analyzable properties Does anyone suggest imaginary numbers exist? Nevertheless, they appear in physics equations.
I don't think it's Platonism because it assumes an independent existence outside of consciousness. The triangulation of a roof truss exists abstractly. The connection can be 'severed' and independent of the concrete thing itself, the roof truss.
Quoting Relativist
Yes, what you stated is the phenomenon relative to metaphyscal language.
Quoting Relativist
Yes, I agree. But they [abstracts] are not needed to build concrete things, that exist.
Agreed. The question remains: do immaterial objects exist? If so, what does it mean to exist? Does Spider-Man exist? Do all fictions, past present, and future exist? What about possible fictions that never get authored?
Fair point, although infinites appear in some physics equations, and they are treated ad objects in transfinite math. Regardless, from this viewpoint, the question is: is there something that exists in the real world that maps to an infinite set?
Think of time itself, as being both abstract and concrete. (Does time exist? And how does it exist, abstractly?)
The presence of such questions doesn't impugn existential quantification.
You want mathematics not to claim that there exist infinite sets; you want to "reject existence", as I understood you, such as existence asserted with the existential quantifer. But you haven't answered my point that without infinitistic set theory, axiomatizing the mathematics for the sciences gets a lot more complicated.
Quoting 3017amen
OK, but that's just referring to a concept - a mental object. It is spatially located in your brain, unless dualism is true. Triangular objects exist even if there are no minds to conceptualize triangles. When people speak of the existence of infinity they are not merely referring to the concept that exists in our minds.
Sorry. I agree with that. They are useful fictions.
No, but one shouldn't conflate existential quantification with a statement of ontology. IOW just because we can do some useful math with infinities doesn't entail anything ontic.
That raises the question, "What do you mean by 'the real world'"? And what do you mean by "something exists in the real world"?
Anyway, whatever the answer about infinite sets, mathematical statements themeselves are things like "There exists the set of natural numbers" or "There exists an x such that, for all y, if y is a natural number then y is an element of x" or "With these axioms we prove the formula 'There exists the set of natural numbers'" and not "There exists the set of natural numbers in the real world (whatever "in the real world" might mean)."
I don't opine on that particular philosophical position. But the outcome of it doesn't preclude existential quantifcation in mathematics or working with infinitistic set theory.
I don't opine whether they are fictions or not. But, for me, at least such views as fictionalism and instrumentalism that allow sets as consistent fictions is enough for working in mathematics.
Yes, and thus we get into metaphysics. A topic for another day.
How is that possible?
What does state of affairs mean?
A triangular object has 3 sides that are arranged in a certain general way. It's existence and structure is not dependent on a mind analyzing that structure.
In a humanistic sense, are you saying that we all are an interconnected consciousness?
Isn't that Platonism?
Not really. The relations between consciousnesses seems indirect.
That's not correct. Platonism involves mind-independent objects. So, how does your triangle exist without consciousness?
Can you explain that better? For example, what does 'indirect' viz consciousness mean (subconsciousness/unconsciousness)?
Mind independent ABSTRACT objects, right?
The abstraction "triangle" that exists in your brain is spatially located in your brain, so it is not the identical object located in my brain.
Then how could we use sense data to apprehend abstract triangles without consciousness?
From a metaphysical standpoint, Charles Sanders Peirce drew a helpful distinction between reality and existence. The real is that which is as it is regardless of what any individual mind or finite group of minds thinks about it, while existence is reaction with other like things in the environment. Accordingly, abstractions and other immaterial objects do not exist, but some of them are nevertheless real.
Quoting Relativist
Metaphysically, Spider-Man and other fictions do not exist and are not real. However, they "exist" in logic where we redefine the scope of that term to a certain universe of discourse, rather than reality itself. Within the fictional world created by certain comic books, movies, etc. there exists an x such that x is Spider-Man; within the real world, no such x exists. On the other hand, those concrete comic books, movies, etc. obviously do exist metaphysically.
Quoting 3017amen
What does it mean to say that time is somehow "both abstract and concrete"? I suggest that time is another example of something that does not exist, but is real. It is a law that governs existents, rather than an existent that reacts with other existents.
Quoting Relativist
I prefer Peirce's definition: "Time is a certain general respect relative to different determinations of which states of things otherwise impossible may be realized. Namely, if P and Q are two logically possible states of things, (abstraction being made of time) but are logically incompossible, they may be realized in respect to different determinations of time." Or as he put it elsewhere: "Time is that diversity of existence whereby that which is existentially a subject is enabled to receive contrary determinations in existence," such that "a real event" is "an existential junction of incompossible facts."
Quoting 3017amen
According to Peirce, "A state of things is an abstract constituent part of reality, of such a nature that a proposition is needed to represent it." Moreover, "A fact is so highly a prescissively abstract state of things, that it can be wholly represented in a simple proposition." As I suggested recently in another thread, every proposition signifies a state of things by attributing abstract characters to concrete things, and every true proposition signifies a fact. Again, the universe of discourse matters--e.g., it is a fact that Spider-Man wears a mask and shoots webs within the fictional Marvel world.
I think of the distinction this way:
Time as an abstract can be defined in this same abstract/paradoxical way. What we perceive we perceive as a present event-as going on now. However, when we perceive such an event, that event is no longer in the present. The actual 'present' exists only for an instant; the event becomes memory. It follows that we perceive the past to experience the present. In fact, it could be argued that the past and future truly exist, while the present is only a variable instant. To me, that whole definition/explaination is an abstract.
Time as concrete is simpler; lunar cycles, harvest season, the sundial, the clock, stop watch, etc.. It's interesting to note that the duration of time has seemingly decreased throughout history.
I think time and eternity can be seen as one in the same.
Peirce makes another helpful distinction between an event as a definite and entire change vs. a state of change as "indefinitely gradual"; i.e., strictly continuous. What they have in common is that both are realized only at a general determination of time (lapse), rather than at an individual determination of time (instant). For any event, the states of things at the commencement and completion of the corresponding lapse are "incompossible facts"; but during the lapse itself, neither of them is realized. The present is always just such a state of change, because various events are constantly in progress throughout the universe.
Quoting 3017amen
I suggest instead that we directly perceive the event as it happens in the present, which again is an indefinite lapse (not a distinct instant) during which a state of change is realized. However, we then involuntarily make a perceptual judgment about the event, and this is indeed in retrospect; so all our knowledge is about the past. Peirce even defines the past as "that part of time with which memory is concerned" and the future as "that part of time with which the will is concerned," such that "Events past are recalled by memory supposing they acted on our sense; events to come are anticipated supposing they are subject to our will."
Quoting 3017amen
On the contrary, using Peirce's definition, the past truly exists--it acts on us, and we react to it--but the future does not. The past is determinate, while the future is indeterminate; so the present "is plainly that Nascent State between the Determinate and the Indeterminate." Likewise, the past is actuality, while the future is possibility and (conditional) necessity; so the present "is the Nascent State of the Actual."
Quoting 3017amen
These are not real parts of time itself, but rather arbitrary intervals between states of things that are similar and regular enough for us to use them conveniently to mark and measure the passage of time.
Quoting 3017amen
According to whom? What exactly does "duration" mean in this context?
Please cite a legitimate reference. Is this claim related to the observation that time seems to pass more rapidly as we grow older? A psychological phenomenon.
To terminate this discussion let me reveal that on my desk is a small Egyptian box, 5cmX3cmX1cm, in which I keep both zero and infinity. So you have my word that, yes, they do exist. :cool:
Hey jgill!
It's really common sense. Clocks are just the way we take the abstract and make it concrete. In this case, a clock serves as a metering and measuring device that gauges change.
The other common sense part is based on history. From ancient sun dials to early Americans harvest season, to modern day atomic clocks, time has in effect, been shortened in it's perception of change.
Relativistic physics talks about "time" speeding up or slowing down depending on how fast you travel. All that changes is the speed of change from the object that travels faster or slower.
But yeah in an anecdotal way I've often wondered about that psychological phenomenon too, as to why time seems to go by faster as we age. Go figure :chin:
The clock itself is concrete, but the time that it marks and measures is still abstract, and the units by which it marks and measures time are arbitrarily defined.
Quoting 3017amen
So the duration of time itself has not decreased throughout history, only the smallest measurable unit of time; i.e., we can mark and measure time more precisely than our ancestors.
Quoting 3017amen
My longstanding hypothesis about this is that as each day passes, it becomes a smaller and smaller fraction of one's entire lifespan. One year out of ten is a sizable chunk, but one year out of fifty--not so much.
Hypothetically, suppose there are 2 "universes" (A and B), the products of independent big bangs, but separated by sufficient space (or dimensionally) that they are causally independent of one another. From A's perspective, does B exist? Is B real?
I am inclined to say no, since there is nothing in B that reacts with anything in A.
Quoting Relativist
I am inclined to say yes, since B is as it is regardless of what anyone thinks about it.
Yep, agreed!!
Quoting aletheist
Yep, but don't forget, as I suggested, it's the perception of it, that has changed. Of course, that begs other questions relative to sense data viz consciousness... . In other words, time appears to be just an illusion. Yet another mystery in life!
Quoting aletheist
Ha, yeah. I often thought of it that way too, only kind of like a cone that narrows. However, that cone that is narrowing, must indeed either be a subconscious/conscious phenomena.., or perhaps the quality of one's life gets better with time thus making time seem to go by faster... .
No, it simply maintains the definition of existence as reaction with other things in the environment. Existence is a special kind of reality, which is a special kind of being. Everything that exists is real, but there are realities that do not exist (e.g., some abstractions); and everything that is real has being, but there are beings that are not real (e.g., fictions).
The point is that what exists for us is whatever can react with us. If it is utterly impossible for B to react with us, then we have no basis for claiming that B exists.
I take no exception to Peirce's view or philosophy there (particularly since he is a close relative of James-pragmatism, ha). However, I'm not totally on-board with his esoteric definition of metaphysics...but like his thoughts and connection to the Will there. Thanks for sharing.
What is your take on his Objective Idealism?
Quoting aletheist
Not sure I agree with him there. How is that so? Is he saying that it is not reasonable, through induction, that the sun will rise tomorrow?
Quoting aletheist
Yep, agreed!
Quoting aletheist
I covered that in the previous post, I think, thanks again aletheist for sharing.
I am not sure what you mean by "his esoteric definition of metaphysics." For Peirce, "Metaphysics consists in the results of the absolute acceptance of logical principles not merely as regulatively valid, but as truths of being." Accordingly, "Just as the logical verb with its signification reappears in metaphysics as a Quality, an ens having a Nature as its mode of being, and as a logical individual subject reappears in metaphysics as a Thing, an ens having Existence as its mode of being, so the logical reason, or premise, reappears in metaphysics as a Reason, an ens having a Reality, consisting in a ruling both of the outward and of the inward world, as its mode of being. The being of the quality lies wholly in itself, the being of the thing lies in opposition to other things, the being of the reason lies in its bringing qualities and things together." These three modes of being correspond respectively to abstract characters as denoted by general terms, concrete things as denoted by quantified variables, and prescissive facts as signified by propositions.
Quoting 3017amen
I find it persuasive. "The one intelligible theory of the universe is that of objective idealism, that matter is effete mind, inveterate habits becoming physical laws," such that "what we call matter is not completely dead, but is merely mind hidebound with habits." Physical motion is thus a degenerate form of psychical thought or semeiosis. "Just as it is strictly correct to say that nobody is ever in an exact Position (except instantaneously, and an Instant is a fiction, or ens rationis), but Positions are either vaguely described states of motion of small range, or else (what is the better view), are entia rationis (i.e. fictions recognized to be fictions, and thus no longer fictions) invented for the purposes of closer descriptions of states of motion; so likewise, Thought (I am not talking Psychology, but Logic, or the essence of Semeiotics) cannot, from the nature of it, be at rest, or be anything but inferential process; and propositions are either roughly described states of Thought-motion, or are artificial creations intended to render the description of Thought-motion possible; and Names are creations of a second order serving to render the representation of propositions possible."
Quoting 3017amen
No, it is a reasonable but fallible prediction based on our knowledge of the real laws that govern existents. Propositions about the future are more appropriately stated as subjunctive conditionals--if the relevant circumstances were to remain unchanged, then the sun would rise tomorrow.
Thanks aletheist. Okay, but with respect to infinity, how does that square with the common paradox of past present and future? I thought you said that he denied future tense.
Or maybe he's thinking that everything is a subjective illusion... which on the surface I would have no quarrel with, viz the abstracts of time.
Again, I am not sure what exactly you have in mind here. How would you succinctly summarize "the common paradox of past present and future"?
Quoting 3017amen
I said that the future does not exist because it is indeterminate; i.e., nothing in the future is actual, it is either possible or (conditionally) necessary, which are two sides of the same modal coin. That is why I rephrased your declarative proposition in future tense as a subjunctive conditional. Among the popular modern theories of time, Peirce's view seems closest to the "growing block universe."
Quoting 3017amen
Definitely not; again, there are realities which are as they are regardless of what any individual mind or finite group of minds thinks about it.
Okay, that's what I thought. And so it begs the question of why. Meaning, in the context of the paradoxical concept of time/infinity, why does he make past and future tense mutually exclusive? He's embracing past tense as reality, yet denies future tense as reality, right?
Simple example: If someone on the west coast is making a call to someone on the east coast, are they not talking to someone in the future? What would Peirce say?
Sorry for all the questions, I'm sure I'm just missing something obvious there...
There are components of the future that are determinate, for example: the positions of the planets with respect to the sun, at this exact time tomorrow. Does this determinacy mean it exists?
No, this conflates reality with existence. The past exists, because it is determinate; the future does not (yet) exist, because it is indeterminate. However, there are real possibilities and real (conditional) necessities in the future, just no real actualities. The present is when some of those future possibilities and necessities become past actualities. The "arrow of time" reflects how the universe is proceeding from an ideal state of complete indeterminacy at the hypothetical commencement of all time, when everything would have been in the future, toward an ideal state of complete determinacy at the hypothetical completion of all time, when everything would be in the past.
Quoting 3017amen
No, both of them are talking to each other in the present. The difference in their spatial locations has no bearing on their temporal relation. The fact that an east coast clock reads three hours later than a west coast clock is an arbitrary convention of how we mark and measure time, and reflects nothing about the real nature of time itself.
Ah, this is where Peirce is wrong. In theoretical physics, determinism/indeterminism relates to causation, not time or infinity, as in our case. The common example of indeterminacy there is the Heisenberg Uncertainty principle.
Quoting aletheist
Well, if that were true, then in reality, one would not be able to call anyone in any other time zone. But since we are capable of such, Peirce's philosophy appears flawed.
And so, all you can say there is that the past and future are, once again, abstract illusions. But the paradox presents itself when one tries, like apparently Pierce has tried, to deny any one of them. He's trying to make the reality of time and its tenses mutually exclusive.
He should accept both as abstracts, but more importantly, much like in the video (I had posted), past present and future is just an abstract. In other words, he should accept reality of future tense, but can relegate it to an abstract illusion.
Peirce's view was that nothing in the future is strictly determinate; i.e., he rejected determinism, which he usually called necessitarianism, instead embracing the reality of absolute chance. Again, it is better to say that if the relevant circumstances were to remain unchanged, then the planets would appear in certain positions with respect to the sun at a specified time tomorrow.
No, this is a category mistake. We are talking about philosophy--specifically, logic and metaphysics--not theoretical physics.
Quoting 3017amen
Nonsense, time zones are arbitrary human constructs for marking and measuring time. The east coast is not three hours in the future relative to the west coast.
Quoting 3017amen
I have no idea what you mean by any of this. If all past actualities exist, and some future possibilities and necessities are real, then neither past nor future is an abstract illusion. Again, what specific paradox do you have in mind? How is Peirce denying the past or the future? On the contrary, he is simply recognizing how they are different. If you want to insist that they are exactly the same, then we obviously disagree about that.
Quoting 3017amen
If something is an abstract illusion, then by definition it is not a reality.
On the contrary, Peirce consistently affirmed the reality of laws of nature, although he held that they have evolved and are still evolving; and again, due to the reality of absolute chance, they are not strictly exceptionless.
I think you're adding a bit to the confusion. Relativist and myself are confused with your philosophy regarding what is natural and an illusion.
You are contradicting yourself when you say Pierce recognizes the laws of nature but when we talk about physics you are denying such phenomena.
You also seem to be denying the paradox between what's reality and illusion
Quoting aletheist
That's correct. The reality of time is just an illusion. And therein lies your paradox.
Where have I ever said anything about "what is natural and an illusion"? I have consistently been discussing the distinction between existence and reality.
Quoting 3017amen
Where have I ever denied laws of nature? On the contrary, I have explicitly affirmed them, but suggested that they are not strictly exceptionless due to the reality of absolute chance.
Quoting 3017amen
I keep asking you to provide a succinct summary of the specific paradox that you have in mind, and you keep failing to do so.
Quoting 3017amen
No, either time is a reality or time is an illusion. I hold that time is a reality, and you apparently hold that time is an illusion; so there is no paradox, we simply disagree.
If you care to continue the discussion, let's start with this basic understanding regarding time as an illusion, and you tell me what is wrong with this paradox(s):
Sorry, I hate watching videos and strongly prefer reading, so I will ask one more time: Please provide a succinct summary of the specific paradox that you have in mind.
Since I do value the conversation, I went ahead and watched the video more carefully, this time typing up a transcription so that I could study what was said and provide my own summary.
The first alleged paradox (attributed to Aristotle) is that the past does not exist (although it did exist), the future does not exist (although it will exist), and the present is just a limit between them; "so time seems to be a nothing dividing something nonexistent from something nonexistent." As I have already explained, my view (and Peirce's) is that the past does exist, and the present--as the determination of time at which the nonexistent (possible/necessary) future becomes the existent (actual) past--is an indefinite lapse, rather than a durationless instant.
The second alleged paradox (attributed to McTaggart) is not as clearly identified. One candidate is that "every event must be one or the other [of past/present/future], but no event can be more than one." My view (and Peirce's) is that every event is realized at a lapse of time such that it is future before the commencement of the lapse, when the initial state of things is realized; present during the lapse itself, when a gradual state of change is realized; and past after the completion of the lapse, when the new state of things is realized.
The other candidate is that "this moment is the only present moment; on the other hand, every moment is present." This is straightforwardly false, or at best equivocal. Every moment is present when it is present, but no longer present when it is past and not yet present when it is future. Moreover, the continuity of time is such that being present rather than past or future is a matter of degree, not a sharp distinction. In Peirce's words, "The events of a day are less mediately present to the mind than the events of a year; the events of a second less mediately present than the events of a day."
Hence I continue to disagree with you (and McTaggart) that time is an unreal illusion.
Yes.
As far as we can see integer numbers are instantiated or mapped to the real world, how do you find it reasonable to assume this relation abruptly stops above some very large number or below number one?
I was referring to that fact that infinity is not a number that is mapped to. That fact doesn't entail an upper bound.
And yet, if space is finite, it's contents are finite - which would entail some upper bound. Current physics indicates that space, and its contents, extends in space through a temporal process (described here). This means the extent can only be unbounded if the past is unbounded (which the article also states). The article indicates that current physics does not establish whether there was, or was not, a past boundary in time.
However, there are hypotheses that do establish a past boundary in time. For example, Stephen Hawking's last paper. There are others that entail a finite past, and none that establish an infinite past - they only fail to establish a physical basis for a temporal boundary.
So there may be a physical basis for a past temporal boundary, and there can be no physical basis for an eternal past. But there is a conceptual problem with an infinite past: completeness. The universe, as it exists at this moment (its current extent, including everything within it) reflects a completed temporal process. Processes of infinitely long duration entail incompleteness at every temporal step. We conceptualize an infinite future as follows: every future point in time, Ti, will be succeeded by a point in time Ti +1. i.e. there is no point in time at which there is completeness. Now turn this around: to reach the current moment would entail a completed infinity, which contradicts this conceptualization. This provides a reasonable conceptual basis for believing the past is finite. This doesn't show it to be logically impossible, but a denial entails a burden to provide a conceptual basis for an infinite (completed) past.
While it is true that abstract concepts only exist in minds as mental entities, minds themselves have no other source of information than the actuality of the external world, so they are all ultimately grounded or abstracted from the real world and are really only extrapolations and variations on the theme provided by the universe itself.
In more poetic language you could say that all we do or think can only be just a reflection of what the universe does and what the universe is. To put it bluntly, a bunch of abstract, meaningless symbols like: 0 = -x + x, can actually have a true metaphysical implication and might be our only way to answer the hardest questions of all, such as “why is there something rather than nothing”, and all about continuity / discreteness and infinity of both time and space.
Indeed the Way of Abstraction is grounded in the real world: first order abstractions are mental creations formed by considering several similar (actually existing objects) and omitting all features except for those held in common. Second order abstractions are formed by extrapolating from first order abstractions - they are abstractions of abstractions, and these are not grounded in existing objects (unless they can also be formed in first-order fashion).
I have an SB in Philosophy. I never studied Philosophy of math. (Though I did take a class on Paradox and Infinity taught by George Boolos.)
In my education, I was always taught that numbers are real, abstract entities. When this was asserted, the professor would typically insert some caveat about how not everyone views numbers in this way, but it was easier to conduct philosophy if you made this assumption. So please let's "bracket" skeptical worries about this, they would say.
I'm sure that when I wrote a paper, I could have adopted either position as a premise without argumentation, and received no red marks, as long as I was consistent wrt to this in my argument.
Personally, I see no good reason to reject numbers as being real, abstract things. In fact, it only seems to make one's life more difficult. There are important questions in Philosophy that I actually lose sleep over, and I don't see how being an anti-realist about numbers is going to help me answers those questions. In fact, it would only seem to make my life more difficult in that regard.
If I'm going to be an anti-realist about numbers, why not also be an anti-realist about chairs, for instance. There are physical things that correspond to our human conception of chairs, but are those things really chairs? Or are they really only just bits of space-time quantum probability wave configurations that we have arbitrarily decided to lump together in a way that does not amount to chairs being "real".
I suppose if I were a "real" philosopher, I might be interested in the intricate minutiae of all the myriad positions and arguments that one can make about exactly it means to be "real". But personally, I guess I'm just a dilettante who is more concerned with the answers to very important questions, like,
(1) "Will that internal Star Trek transporter leave me alive? Or will being transported kill me?" (Wanting to know the answer to this question is actually why I started studying Philosophy, and on the first day of Philosophy 101, Judith Jarvis Thompson pointed at me at the beginning of the class, and asked me why I was here in a Philosophy 101 class. I was too embarrassed by the reason I was there to answer her. Little did I know at the time that this is actually a topic of serious philosophical debate!)
(2) Are p-zombies possible, and if not, what explains the fact that no amount of perfectly good reasoning can convince me that they are not? Well, maybe this really means that interactionism is true and p-zombies can't exist, but interactionism just seems way too problematic.
(3) Why is boiling live babies to death something that I should not do? It seems like an indisputable fact that this is true, but all roads of argumentation that would lead to this conclusion seem to be based on very tenuous premises.
(4) Why is there something rather than nothing? It seems to be incredible that there's all this stuff, rather than nothing at all. (Except for numbers and the like.)
So, all in all, how would rejecting the reality of numbers help me in answering what is really important?
|>ouglas
I’m not sure if that article makes one interesting distinction about the abstraction algorithm called “visual perception”. There are colors, clearly an abstraction of who knows what order, but then, there are lines and shapes, and countable discrete things, not quite abstracted, but rather mapped kind of directly.
That sort of thing hints that geometry may hold the truth and perhaps even all the answers, but unfortunately along also comes everything else with all the non-answers included.
1) the Star Trek transporter will definitely kill you, and then create a replica.
2) p-zombies are possible (I think)
3) objective moral values are not existents, so you need to explore psychology, sociology and evolution to understand why most of us don't want to boil babies alive (btw, I prefer my veal to be grilled).
4) existence (rather than nothingness) may be a brute fact, which may not be satisfying - but does that justify choosing an answer (e.g. "God") just because it doesn't leave you hanging?
The ontological status of abstractions is pertinent to physicalism, because they aren't physical, but abstractions don't actually falsify physicalism because they can be accounted for in the Aristotelian way, as I'm doing: they exist in their instantiations, but do not have independent existence apart from these other than as mental objects.
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Setting aside the question of what quailia are, colors are first order - we consider objects that are red, and abstract redness from the memory of those perceptions. The geometrical figure "line" is second order - we may envision a drawing of a line on a page, which is imperfectly straight, and 3-dimensional, and imagine its ideal form as one dimensional. A similar process with 2 dimensional objects.
We form concepts about countable discrete things and extrapolate, but these things can also be abstracted directly from something discrete.
The article says quite clearly, and more than once, that we don't know whether the universe is spatially finite or infinite. We can estimate a lower bound on its size, but not an upper bound. The simplest topology consistent with our observations at large scales is an infinite, flat space; this is what the most common current cosmological model posits (so-called FLRW model). However, there are also closed topologies that are consistent with the same observations.
As for your conceptual anti-infinitist argument, this is an old and surprisingly persistent confusion. Quentin Smith had a nice analysis of this and several other such arguments in a 1987 paper Infinity and the Past.
Involving infinity neither side of the argument can be semantically valid. One side wants to count from the beginning where there is no beginning, and the other side has to say something like “infinite amount of events actually occurred”.
Infinite events could not have occurred, semantically, they would have to be still occuring, just like infinite future events can not ever occur by some point in future time and instead would have to still be occurring at every point in time, by definition.
Infinity is a concept defining indefinite, there is no reason to expect any logical / semantic manipulation of such a concept can squeeze some definite or meaningful answer out of it. We need to be asking other kinds of questions, it seems, if there is any actual answer to be found at all.
If one apple exists on a table there is one apple. If zero apples exist on a table there are zero apples. zero exists.
As for infinite, there are infinite real numbers and only time will tell whether infinite exists. How much do you want to bet that infinite exists? A penny?
Novel notion, Z! How would you describe 50% of infinity? :chin:
Physics will never be able to prove the past is infinite, all it can possibly do is to show that no past boundary of time has been found.
Thanks for the article. I'd seen it a few years ago, but forgot about it. However, it does not address my argument.
My argument is in the spirit of David Conway’s, in that I utilize the concept of completeness. However, Smith’s refutation doesn’t apply to my argument. I’m not making the bold claim that an infinite past is logically impossible, I simply claim that there’s no conceptual basis for considering it POSSIBLE, and therefore it’s more rational to reject it. An infinite past entails the completion of a sequential process of infinitely many steps, each of which is of finite duration. How can an infinity of days become completed? Our concept of an infinite future entails a process that never ends. This concept isn’t reversible to the past because the past has ended. All Smith does is to assert the past can be mapped to the (infinite) set of negative numbers – a logical relation, that doesn’t account for the process.
Only in your mind, because you're considering the possible presence of apples on the table. Suppose there were oranges and bananas on the table. The negative fact (there are no apples on the table) provides no information about what DOES exist on the table.
There are no more truths than those entailed by the conjunction of all positive truths, so negative truths are redundant.
I'm not insisting that only physical objects have existence - I'm open to other possibilities, but I suggest we should be parsimonious in our assumptions of what actually exists in the world. I'd be fine accepting the existence of angels and devils despite being immaterial, if their existence is needed to explain some aspect of the world. I accept the existence of mental objects (exactly what they are depends on what the nature of mind is). On the other hand, abstract objects (all of them, not just infinity and zero) ostensibly exist independently of minds. Where are they? Why include them an an ontology? They aren't causally efficacious, and they can be accounted for without assuming they are components of the world. We need to treat them as existing when doing math, but this utility doesn't force us to treat them as actual, independent components of the world. Math works just fine even if they're just useful fictions.
Infinity exists for the simple reason that if you were to task me to write down all the natural numbers then that would be an instantiation of infinity in the real world.
If I had 5 pennies in my wallet and I gave you all of them then my empty wallet is a real-world instantiation of zero.
As I see it, the concrete precedes the abstract, the latter being derived from the former. Doesn't it follow then that "all" abstract objects will invariably be instantiated in a concrete object?
It is no more ‘located in the brain’ than actors are located inside televisions. Rather a rational mind is able to recognize such concepts which however are not dependent on being recognized in order to be real.
I think an objection can be raised here at the outset. The statement about 'the real world' falls for what has been described as the 'myth of the given' - which is, as I understand it, the notion that there is indeed a real world which we know independently of the conceptual processes required for the statement 'there is a real world'. In other words, in respect of this topic, it begs the question, i.e. assumes what it sets out to prove.
I say that such an assertion implicitly pre-supposes a division between the 'real world' which is presumed to exist independently of any conceptual framework, and the purported 'internal world' of ideas, concepts and abstractions, but that this division is really a false dichotomy.
The problem is that as soon as you proffer any judgement of what the real world is, then you're already relying on abstractions. You have your ramified concepts of 'what is real' and how to arrive at judgements about it, and so on - all of which rely on abstractions in some fundamental sense, us being language-using beings. Of course, we don't notice we're doing that, and often it doesn't matter that we don't notice it, but in this case, 'noticing it' is fundamental to the topic itself. So at the outset, I'm questioning this assumed distinction between abstracta and purportedly real objects. Humans look at real objects through a framework of abstracta, such as language, mathematics, and scientific theory. What things are outside of, or apart from, that framework, is in some sense a meaningless question. An oak tree is a perch to a bird, a meal to a termite, and fuel to a fire. To a botanist it is a member of the genus Quercus. What is it, really?
And the problem with your theory of numbers, is that it fails to account for the 'unreasonable efficacy of mathematics in the natural sciences'. This unreasonable efficacy has enabled science to perceive many fundamental things which it could never perceive by unaided sensory experience; because, in some sense, mathematical reasoning both predicts and explains relationships on some fundamental level, then science has been able to deduce mathematical principles with universal applicability. And the empiricist dodge of trying to account for them in psychologistic terms doesn't do justice to this ability.
Real numbers, for example, are real in that they're the same for anyone who can count. The nature of their reality is what is at issue - which is an inconvenient truth for naturalism, which believes that existence has a univocal meaning - that everything that exists, exists in the same way. As soon as you're obliged to say ,'well, it depends on what we mean by "exists"', then you're no longer in the territory of naturalism, and many easy assumptions no longer hold.
If there are 2 apples on the table, but not 0 or 3 apples on the table then 0 and 3 still exist. What can be applied to 3 can also be applied to 0 atleast in this case.
Negative facts do not establish what exists.
Quoting TheMadFool
Quoting TheMadFool
We can abstractly consider geometrical objects of 4 or more dimensions. That doesn't imply such things exist in the world.
You are referring to the concept of infinity that can not actually exist, by definition. The concept does not represent a thing to exist or not exist, nor it represents a collection or set of things. It represents an unfinished process of counting, and as such whatever it is accounting for can not exist because it is in perpetual state of becoming, just like you can not say a washing machine exists while its parts are still on assembly line and has not been put together yet.
Each TV has its own set of pixel producing devices, and while you and I may perceive nearly identical images, the images in my brain are in MY brain, not yours.
Abstactions like triangles are well defined, that's why we can each consider objects with triangular properties, and engage in the way of abstraction. That does not entail the independent existence of the abstraction, "triangle".
It is located in the brain just like simulated crocodile is located inside the computer and just like representations of actors are located inside a TV set. It's a virtual or mental kind of existence, a form of abstraction. i.e. representation.
A washing machine is assembled in ordered steps. If these are known then each step is a stage in the construction of the washing machine. Similarly the simple process of adding 1 to a preceding number as in 0,1, 2, 3,... is a buildup to infinity.
Process describing a thing that will never be completed, that will never exist.
Yes, I assume that the images in our mind are different from the objects beyond our minds. Only you have access to your mind's contents; the content is subjective, and it comes to exist solely through mental activity, and ceases to exist when your mind ceases to exist (or sooner; you don't remember all the details of all your past experiences). This is different from objects of the external world, which exist independent of minds - they have objective existence.
Quoting Wayfarer
The objects of the world are states of affairs whose constituents have relations to one another, and these relations can be described mathematically. I'm not saying the rekations don't exist, I'm just saying they don't exist independently of the states of affairs in which they are instantiated.
Does it really make more sense to suggest the relations exist independently, and the states of affairs that exhibit them have some sort of ontic relation to those platonic objects? The efficacy of math is not dependent on it.
I was talking about spatial extension. Simply put, if space is infinite, which seems plausible from what we know, and if the rest of it looks much like what we can see around us, which is very plausible, then there's your actual infinity (if by that you mean an infinite number of material objects).
Quoting Relativist
Quoting Relativist
You do not so much utilize the concept of completeness as just plug it in and expect it to do the work for you. Try to unpack the reasoning and you will see that it either does not apply or it begs the question against the existence of actual infinities.
Smith does address the sort of argument that you are hinting at in his section VI:
(And he goes on to address Conway's and Craig's arguments in that vein.)
Quoting Relativist
I fail to see the distinction that you are trying to draw here.
Your suggesting that: since concrete objects entail abstract objects, that all abstract objects entail concrete objects. That does not follow; it commits the fallacy of affirming the consequent.
First order abstractions are formed by considering multiple objects of the world that have some common features, and mentally discarding the features that distinguish them. Second order abstractions are mentally constructed by extrapolation of first order absteactions - they don't necessarily have instantiations in the real world. Consider extrapolating from squares to cubes, to tesseracts, and beyond to higher dimensional analogues of cubes. They can only exist in the world if the world actually has that number of spatial dimensions.
The first article showed that, according to accepted physics, spatial extension can only be infinite if there is an infinite past - that's why I focused on past time.
Quoting SophistiCat
This does not apply to my claim. Sure, it's coherent - his statement entails no logical contradiction, but it circularly assumes the infinity exists, and it is that assumption that I challenge. The concept of infinity is a second order abstraction - an extrapolation of first order abstractions. (see the last paragraph of my last response to TheMadFool). A 90-dimensional analogue of a cube is an extrapolation of a cube. It could be described coherently, and consistent mathematical inferences could be made- nevertheless, this does not justify believing there exist objects of the world that correspond to it (there would actually have to exist 90 spatial dimensions). The general lesson is that we should be suspicious of second order abstractions - the mere fact that they have logicallly coherent properties does not establish their having real-world instantiations. Something more than logical coherence is needed to justify believing it.
I hope you see that my argument is purely epistemological, and specifically concerns whether or not belief in an actual infinity is justifiable.
If you mean this article, then it doesn't say that, and it would be pretty incredible if it did, because the best established cosmological model to date implies an infinitely extended universe without the assumption of an infinite past. Note that the article spends considerable time talking about inflation, which is less well established than standard Big Bang cosmology, but inflation doesn't imply what you claim either.
Quoting Relativist
There is no circularity, because the refutation does not seek to establish the positive claim and makes no assumptions of its own. It only shows that the anti-infinitist argument doesn't work - the specific argument that you were making originally involving "completeness." (Morriston makes a similar case in Must the Past have a Beginning?)
As for your first order/second order abstractions, that's something completely different. I won't go further than just to say that I am not buying your epistemological construction.
The article states: "In fact, unless inflation went on for a truly infinite amount of time, or the Universe was born infinitely large, the Universe ought to be finite in extent." I admit I was only addressing the first possibility, but the second possibility remains just a brute fact assumption.
Physics analysis can't apply an arbitrary limit, so they must start with an absence of boundaries. That does not establish the existence of an actual infinity, it just indicates that the physics entails no boundary.
Quoting SophistiCat
At least you see it's different. I consider rational belief to require rational justification, and logical consistency seems inadequate as a rational justification. 90-dimensional cube analogues are logically consistent, but there's no rational basis for believing they exist in objective reality.
Infinity is not a number. No logically valid statement can account for uncountable.
Starts with fallacy, continues with assumption, ends with paradox.
As is the alternative. It is widely understood that the universe (or at least the inflationary "pocket" that we occupy, if we go with the still somewhat speculative eternal inflation hypothesis) was either always finite or always infinite. Neither option is granted the default status by cosmologists (your epistemological arguments notwithstanding). The question is decided by the weight of evidence, or lacking that, by considerations of simplicity; these two epistemological criteria constitute the foundation of scientific epistemology. As far as the former, there have been some publications that claimed that a closed (finite) spacetime topology fits some of the observational evidence better than the flat topology of the standard FLRW model, but their conclusions have not yet gained widespread acceptance. As for simplicity, here the flat semi-Euclidean spacetime wins over the more structurally complex closed topologies.
But what is the status of the definition? I would have thought that triangles would be discovered in all possible worlds, in other words, their reality is not dependent on our definition of it, but our definition of it must conform to the concept (of a flat plane bounded by three intersecting straight lines). The issue is that your notion of 'existence' is too narrow; it comprises only physical objects, actual triangles, which you assume are alone 'real objects'. But geometrical concepts, such as triangles, are first and foremost concepts, of which physical forms are but representations. So rather than triangles being physical objects represented by ideas in the mind, the idea 'triangle' is a concept represented by physical examples.
Quoting Relativist
We know of anything beyond the mind through the processes of assimilation and synthesis, which takes place in the mind. We have a picture of the world, a master-construct if you like, of the domain of objects with you, the subject, within it. But this doesn't acknowledge the sense in which this construction of self-and-world is also the work of the mind (referred to in, for example, Schopenhauer's philosophy as 'vorstellung'. ) So again, in everything you write, you're assuming a reality independent of any observing mind, but that assumption might be philosophically problematical, i.e. it might not be as self-evident as you're assuming it to be.
Quoting Relativist
I believe so. Don't overlook the fact that the current model of the so-called fundamental constituents of matter, the so-called 'particle zoo', is a mathematical model. Certainly it has to be validated against empirical data, but it is arguably a purely intelligible construct (which is close to what some physicists say about it.) Have a look at this paragraph from the SEP entry on mathematical platonism.
I see. If I understood you correctly then there are such things as second-order abstract objects e.g. tesseract. However zero and infinity are first-order abstractions in my opinion; so they should have concrete instantiations.
Why do you say that? A process that has begun represents the process right?
Neither the process nor that it has begun was focus of my point, but infinite set the process is assembling and the fact that it can never be complete, by definition. Infinite set will never be complete, therefore infinite set will never exist. There is no such thing as infinite past, it's simply a self-contradiction.
Of course it can't be completed but it can be started. Look: 0, 1, 2, 3, 4, 5,...
Are you a robot?
Infinite set will never be complete, therefore infinite set will never exist.
Okay. I still think when you do the simple phone call across the globe and/or 4th dimensional time travel, it creates the paradox from present tense. For instance, when traveling from west to east, another paradox presents itself by virtue of one being unable to live the lost hours (or in the opposite you get to relive them).
Nonetheless, the video synopsis could be summed up in the simple statement:
1. Eternity is Time. Time, eternity.
Is that true or false?
(Some things we covered)
a. it's an abstract
b. it's a reality
c. it's an illusion
d. it's all of the above
e. it's something else
If time is illusion how do you explain velocity and motion in general?
Hi Zelebg!
Generally, you could explain it through a simple mathematical formula (time= distance/speed). And as such, we are back to abstracts. Mathematical abstracts.
Are they real? (Where do they come from, a priori.) How is this phenomenon even possible?
These are intriguing questions for sure... !
Zero is second order: it is conceived as a negative fact, like removing the apples from a basket, one by one, ultimately leaving 0 apples. Negative facts are indirect - they don't tell us what IS, they tell us a subset of what isn't. One refers to 0 apples in this example only because of the psychological context - we're considering apples. Although the basket has no apples, it may contain oranges and orangutans. It is the indirect nature of negative facts that makes them second order.
Quoting TheMadFool
Consider how the notion of infinity is manifested in your counting example. Counting is a process. There is no infinity at any identifiable step of the process. Rather, infinity manifests as the process itself, one that never ends - and process is not an existent. It's still a reasonable way to conceive of a potential infinity, but this conception doesn't work for an actual infinity - including a past infinity. A conception for an actual infinity cannot be some ongoing process. It would entail a COMPLETED process.
speed = distance / time
So now I explained how time is in fact not an illusion. What the hell am I doing here when I have more meaningful conversations with my dog.
Hahaha, in a strange way...kind of reminds me of the pyramids at Giza... we only figured that out, abstractly :brow:
Mathematical models don't entail the independent existence of platonic entities.
Consider the simple case of the relations between electrons and protons. How do we account for these with platonic entities? We'd have an electron with some sort of ontic relation to the abstract object "-1 electric charge", and the proton has an ontic relation to the abstract object "+1 electric charge", and then these abstractions have an "attraction relation" between them. It is simpler to simply assume the attraction relation exists between electrons and protons by virtue of their respective intrinsic properties (their electric charge). This account minimizes the number of ontic entities, and does not constrain our ability to reason and do math.
The definition of triangle is not impacted, just the ontology behind it. There are states of affairs that have triangularity among the constituents. We can mentally consider the property "triangular" while ignoring the other details of the object, but that doesn't require the abstract object "triangle" to be ontic. It's a semantic convenience, and has the utility of allowing us to do the math,but we can do the math witthout making an ontological commitment to them.
Quoting Wayfarer
Yes, I'm assuming a reality independent of the observing mind; i..e. I deny solipsism. But this belief is not a deduction, it's just an articulation of a component of our intrinsic (not deduced) view of the world. While it's possibly false, that mere possibility is not adequate grounds to undercut the belief in an external world - a belief that is directly derived from considering our hard-wired relation to that world.
This is about ontology- what actually exists, and we should not be promiscuous with our assumptions about what exists. Triangularity still exists, and we can still reason with triangles (as well-defined mental objects) without adding them to the furniture of the world.
No, there are no "lost" or "relived" hours. That is an illusion created by our arbitrary manner of marking and measuring time.
Quoting 3017amen
Where are you getting that from the video? For one thing, the word "eternity" is never mentioned.
Quoting 3017amen
I cannot answer until you elaborate on what you mean by it. In any case, I have just started a new thread on "The Reality of Time" and suggest that we continue this conversation over there.
Distance exists in the real world as an ontic relation between two objects (two states of affairs) separated in space. Time exists as an ontic relation between two events (two states of affairs) separated in time. Considered apart from their respective objects, distance and time are abstractions - mental objects, that can be abstractly mapped to numbers, which we can mentally manipulate mathematically - computing such things as ratios (distance/time = speed).
Again, your definition of what exists is too narrow. It's restricted to objects. But mathematical relationships and ratios also exist - it's simply that the don't exist as objects, but are implicit in our understanding of what objects are, and how they relate to each other. So saying that numbers and the like are real, is not saying that they exist as literal objects, but as they're intrinsic to rational thought then they're real in a way that literal objects are not.
Quoting Relativist
I can see we're not going to clear this hurdle, but for what it's worth: in referring to the reality of mind, I'm not referring to your mind, or my mind, or any individual instance of consciousness. I'm referring to the vast expanse of shared meanings within which all discourse about objects is conducted. You think that what 'really exists' are only the objects of perception within this matrix of ideas, but I'm arguing that it's the matrix of ideas that is real. (This is near in meaning to the ancient Greek term 'nous'.)
But the underlying issue is a very deep one: modern science (and so much of modern thought) starts with the methodological assumption of a mind-independent world; which for the purposes of science, is a valid assumption. But it has no ultimate reality, because reality itself is not something that we're outside of, or apart from. This is something that has started to become clear in recent science in multiple ways (see for instance this blog post.)
I disagree with describing it "narrow". It is parsimonious, but leaves nothing unaccounted for. It's reasonable to methodologically treat abstractions as independent existents, but that utility does not depend on an ontological commitment.
Emphasis on the qualifier "independent", because this parsimonious ontology doesn't deny the existence of triangles and right angles, it just denies that they exist independently of the things that have those properties. The angles between the walls of my bedroom are 90 degrees - and this angle does actually exist, just not independently of the walls. Many other things have this exact same property, and that's why "having a 90 degree angle" is a universal.
My issue is that there's no good reason to assume "90 degree angle" exists independent of the things that have it. Sure, we can think abstractly about this property without considering the things that have it, and that's a product of our powers of abstraction.
If you don't accept my premise about parsimony (that we should minimize the ontological furniture), then you are free to assume "90 degree angle" actually is an independently existing Platonic entity. But you would need to account for the relation between this entity and the items that exhibit it.You also need to distinguish between abstractions that actually exist (like "90 degree angle") and fictions that exist only within minds (single minds, or even many minds) - fictions like Spider-Man.
That’s right. The issue is this should be self-evident and the problem is people just don’t understand the words. If they did it would be clear “abstraction” is a form of interpretation or understanding, and as such it requires an interpreter and grounding against any information will be understood.
I can look at a mountain and see a triangle, interpret it in terms of spatial relations and then abstract an idea of an angle from it. Someone else can interpret it as “no parking allowed here”. In any case without an interpreter to form a meaning or an idea, any information is just an arrangement of physical stuff, and there is no triangle out there, there is no angle, no mountain. There is only “stuff”, some of which we choose to isolate or abstract as a separate thing and put a label on it.
On the other hand, there are principles like path of the least resistance, Pythagoras' theorem, or inverse square law that are kind of opposing field / particle physics, in a sense that it is not quite clear what comes first, which laws, principles or physical properties are primary and which just a consequence. This sort of ontological dilemma, in contrast, is worth of discussion.
I think it's a product of the pedagogy of mathematics, physics, logic and some related fields. We're taught that triangles and laws of physics (expressed as equations) exist. This leads us to speak of them that way, and this leads to treating them as ontic, and not just as a manner of speaking.
From the point of view of a mathematician or physicist doing his normal work, it doesn't matter. It's convenient to treat them as existing, as a methodological principle. But when mathematician's and physicists conflate the methodological semantics with ontological claims, they've gone too far.