(\frac{1}{\epsilon}) would be ({\infty}), not infinitesimally smaller than ({\epsilon})…
If you divide 1 by a tiny number, you get a huge number. The smaller the denominator, the larger the result.
But according to hyperreals, you could divide ({\epsilon}) by ({\infty}) and get an infinitely smaller value than ({\epsilon}), and so on, infinitely. So there really isn’t a lid on this whole “infinity/infinitessimal” business either way - as the terms would suggest. That’s why it’s pointless trying to assign a finite symbol to infinity/infinitessimal, because for any precision in what it denotes, it can always be bigger/smaller. Infinity is just a pointless concept for human conception. It can only be referred to in a handwavey “tends towards” way, such that it refers to finites getting really big or small to point us towards a precise finite result - such as with limits and calculus etc.
Exact is a property of perceptual acuity. If we travel 6 miles away from a bolt on the sidewalk we can’t see it, if we use an electron microscope it is no longer a bolt.
Everything, everything in existence is not just the middle ground, it is “good enough”.
Mathematics seems to me to be the thought experiment of “what if precision were possible”. This is where the traditional split occurs between theorists and practitioners - when one prefers the aesthetics of either “what if it were precise” (e.g. math/logic) or “let’s just make it do something” (e.g. experimentalists/engineers). The latter accept “good enough”, the former do not.
There seems to be a clash of the two when it comes to relativity, where multiple perceptions occur depending on the point of reference, yet the mathematics allow a congruent consistency regardless.
I’m also reminded of fractals, where the degree to which you “zoom” into the detail affects a calculation - in the sense that strictly a perimeter might seem to be infinite, but a meaningful finite calculation is also possible when you don’t “zoom” in completely. A famous example of this would be the length of coastlines, or landmass perimeter.
“Identity” in general falls squarely into this dilemma, such as in the example you provide. The struggle to find utility is what humans seem primed to attempt - hence Experientialism, which distinguishes truth from utility. Whether or not it’s up to “perceptual acuity”, as you have coined, appears to be a matter of aesthetics. Mathematically, exact equality is permitted.
It’s an interesting thought experiment, but if things are exactly equal to each other, everything would be exactly the same… which is the same as nothing at all (which isn’t true). It’s all about utilitarian approximation
But that’s not what math does - it apportions things (i.e. rationalisation) such that inequality is everywhere, but proportional to everything else - allowing equality to be calculable via ratios. That’s why e.g. energy isn’t the same as mass, but it’s proportional in relation to the speed of light squared. They’re not exactly the same, but they can be expressed as exactly the same as a ratio of something else. That’s the utility of pure mathematics, as tested to a “utilitarian approximation” when scientific experimentation is performed. The two feed back on each other dialectically. It is indeed an interesting thought experiment for the very reason that its practical approximations really do have utility i.e. “science works”. It’s curious, no? That applied theory, despite its practical approximations when tested, actually works - and consistently too. It’s almost as if life can at least imitate precision, and maybe it’s just our attempts to test it that are the only imprecise element.
What do you mean separated? Certainly, if you believe that what separates 1 from (0.\dot9) is an infinitesimal, then you believe that what separates 2 from (1.\dot9) is also an infinitesimal. Yet they are separate by 1 whole number.
But what do you say about my comment about dividing an infinitesimal by infinity?
What is 999…m and 888…m?
Yikes, indeed! You’re so right, Silhouette. I sometimes make stupid mistaken, and that’s a clear example of one. But you got what I meant: (\frac{\epsilon}{\infty}).
Infinity just means “no end”. The confusion arises when the concept gets objectified (thought of like an object) and people confuse it for a number–the “biggest” number, the number farthest out on the number line–the handwaving you mention is a way of talking about infinity as if it were this number–whatever it is you’re calculating, we say it “tends towards” (i.e. gets closer to) that really big number. But I think a more accurate way of saying it is: it tends in the positive direction on the number line with no end.
I think I have come to the conclusion that nothing can be infinitely long or wide other than Existence Itself. Consider the following:
1.999… and 2.888… and 1.888… all denote the Infinitesimal but from different positions. They are not denoting different infinitesimals as there cannot be more than one Infinitesimal.
888…m and 123123…m and 1000…m all denote the Infinite but from different positions. They are not denoting different infinities as there cannot be more than one Infinity. I believe this to be because of the following:
888…m either denotes the Infinite, or it denotes something that goes on forever. Whilst it is necessarily true that the Infinite does go on forever (it has no end), it is also true that the Infinite is actually Infinite (It had no beginning and has no end). This is descriptive of Existence. Nothing else is such that it has no beginning and no end. If something is able to go on forever, it is only because there is Infinity/Existence for this possibility to at all be possible. But that which had a beginning that goes on forever, cannot be described as being Infinity because even if something goes on forever, it is not necessarily Infinite. At all given points in time that x is going forward, he is encompassed by Existence/Infinity. He has an end that he is surpassing every time endlessly. He is not endless like Existence is endless and he will never surpass Existence.
Things that have a starting beginning that are immortal, or things that have a starting point that go on forever, are not necessarily infinite. Only Existence is necessarily Infinite.
Yes, but that does not mean that there are two different infinitesimals separating them. Consider the atoms in my body. Ultimately speaking, the Infinitesimal separates one atom in my body from another. It separates all atoms in my body from each other just as Existence separates one atom in my body from another. With that beings said, what separates one Infinitesimal from another? What separates one Existence from another?
How do you divide the Infinite or the Infinitesimal? And when you divide them, what is the thing that is separating them? Non-existence? Nothing?
Given what you said in your previous post: “888…m and 123123…m and 1000…m all denote the Infinite but from different positions. They are not denoting different infinities as there cannot be more than one Infinity,” I think I understand your point. There can only be one infinitesimal as an amount. In other words, the amount separating (0.\dot9) and 1 is the same amount separating (1.\dot9) and 2. I think this applies to any quantity. There’s no two number fives. There is one. This is why we call it the number 5 rather than a number 5. I think this applies to pretty much anything metaphycial, because without a position in time and space, what is there to separate two different instances? They are essentially one and the same.
Well mathematically speaking, you don’t need anything separating them. Mathematically, division just refers to how big equal parts are relative to the whole. So 5 / 2 = 2.5 just means each half of 5 is 2.5 in size. But I don’t believe one can divide infinity nor an infinitesimal. I don’t even believe such things exist. This is why I said in my first post “If there is an infinitesimal at all…” If there are infinitesimals, then you can have infinitesimals relative to infinitesimals. This is borne out of the mathematics of hyperreals. Take the hyperreal number R, which is an infinitely big number. In the mathematics of hyperreals, R being infinitely big doesn’t mean there can’t be a number larger than it. It just means it’s an infinite distance away from 0 on the number line. But once there (magically), R turns out to be at a point on the number line like any other number, and you can go beyond it: R+1, R+2, R+3, etc. The same logic applies to infinitesimals: it is not the smallest number possible (in terms of absolute value), but that it requires an infinite number of divisions to derive it. But once derived (magically), it turns out to be at a point on the number line like any other number, and you can divide further, coming yet closer to 0.
The word “infinitesimal” merely means an immeasurably small amount.
Until someone does as James did - declare a standard infinitesimal - there are an infinity of infinitesimal amounts just as there are an infinity of infinite amounts.
And more than that, there are many different incidences of an infinitesimal - such as the infinitesimal just above 2.0 and the infinitesimal above 1.0.
What I see as the serious issue is that the expression “0.999…” implies that there is always an infinitesimal amount between that expression and the definite amount of 1.0. No matter how large that infinitesimal might be, it always exists because the expression directly states that there is always some left with that “…” signification.
But in the new world communism what is “true” is exactly and only whatever the Globalist authoritarian leaders declare is true - and so far that seems to be that “1.0 = 0.999…” regardless of any obligation to logic.
The m I meant meters. Your interpretation of my suggestion that there is only one Infinitesimal is interesting. But I meant it more in a not mainstream mathematical kind of way. I think mainstream maths says infinity plus 1 = infinity. Or infinity minus 1 = infinity. To me you cannot add or take away from infinity because that is like saying you can add or take away from Existence. To me, Existence Is Infinity. So by saying there is only one Infinitesimal, I mean to say that Existence = Infinite = Infinitesimal. Some existing thing (which I call Existence) must separate all other existing things (which I call parts of Existence) from each other. My atoms may be separated by x, x may be separated by y. But the Infinitesimal cannot be separated by anything. It is that which separates the things that It sustains whilst nothing sustains It other than Itself. Existence cannot be separated by anything, whilst It is that which sustains and separates all existing things from each other. Non-existence cannot separate Existence or divide Existence/The Infinite/The Infinitesimal. There cannot be more than one Existence.
Maths may have a different view. But then maths would be logically inconsistent would it not? Would it not imply that either the infinitesimal is non-existent, or that the infinitesimal can be separated by something other than infinitesimal?
I need something to = Existence such that it sustains Itself and everything else. Separates all things from each other, and there is only one of It. If I do not have this, I have a paradox in my belief system. Infinity/Infinitesimal satisfies what I need. I cannot view It as being more than one because I cannot view Existence as being more than one. The Infinity/Existence that is in me, is ‘immeasurable’ because Existence is ‘immeasurable’ regardless of whether it’s inside me or outside of me. Although I do not think that Infinity = immeasurable in the sense that it is not a measure. To me, Infinity is Its own Measure. I can understand it, but I can never measure It or reach Its Measure.
I’m not wholly unfamiliar with James’ “Affectance”. I don’t think I have any objections to it, but I hardly know anything about it. But from what I know, it does not aid me in the following:
Some existing thing x must be such that it is absolutely not divisible because it must divide and separate all things that are not x. It must also unify them by virtue of itself. So for example atom p is separated from atom q ultimately as a result of x. x encompasses both p and q such that p and q can be recognised as distinct from one another as a result of x being x.
If x did not separate p and q, then nothing/non-existence would be separating them. And if x did not encompass them such that both p and q are united in being separated and encompassed by x, then p would have absolutely no existing relation to q. Either p would be absurd or q would be absurd or both would be absurd (depends on which is encompassed/sustained by x).
x cannot be anything other than Truly Infinite. Where x is not Truly Infinite, then no one thing encompasses all things and separates all things from each other.
Between 1.999…9 and 2, there is 1.999…
To me 1.999… Is True Infinity. To me 1.999… Is Existence. Just as to me God/Perfection Is Existence. Different labels/symbols, different ways of approaching It (the semantic), but ultimately the same Thing/Semantical value.
All of that sounds like affectance except for one issue, perhaps merely of wording.
When you say -
“Some existing thing x must be such that it is absolutely not divisible” do you consider that affectance can be divided into separate identified portions even though there is always affectance between those portions? There is nothing between portions of affectance except other affectance. There is no non-affectance anywhere at any time.
But affectance does not come in particles (if that is what you were referring to). Affectance forms particles as clumps, “traffic jams” of free flowing affectance that permeates literally all physical existence - the universe is MADE ENTIRELY of affectance - light, matter, magnetism, electric polarity - the whole bag and baggage.
On the question of whether you can add to or take away from infinity, I think it depends on the context. In a purely mathematical context, I agree that you can’t. When mathematicians say that (\infty + 1 = \infty) or (\infty - 1 = \infty), they are say that adding or subtracting 1 from an infinite number of things doesn’t change the fact that it’s infinite. It’s a statement on the property of being infinite, unlike other mathematical expressions which are statements on the quantity of things. IOW, even though the expressions look the same (i.e. they all look like mathematical expression), what they’re saying is fundamentally different.
In a physical context (I don’t know how you can have an infinite number of things physically, but suppose you could), I don’t see any difficulty adding things or taking things away from an infinite set of things. Still doesn’t change the fact that the set is infinite, but you can certainly add to it or take away from it.
I agree that there cannot be more than one existence. But I failed to see how you equate infinity with infinitesimals. It seems like you mean something different from what I offered (infinitesimals as an amount). If you’re saying infinitesimals separate everything in existence, then it would seem you need as many infinitesimals as there are things to separate (and then again for all the points in space constituting each separation).