Steve Patterson, among many points of objection, is doubtful on the modern concept of infinity (or more strictly the transfinite):
The foundations of modern mathematics are flawed. A logical contradiction is nestled at the very core, and it’s been there for a century.
Of all the controversial ideas I hold, this is the most radical. I disagree with nearly all professional mathematicians, and I think they’ve made an elementary error that most children would discover.
It’s about infinity. I’ve written about infinity here, here, and here, and each article points to the same conclusion:
There are no infinite sets.
Not only do infinite sets not exist, but the very concept is logically contradictory – no different than “square circles”.
Infinite sets are quite literally enshrined into the modern foundations of math – with what’s called “The Axiom of Infinity”. It simply states that, “At least one infinite set exists.” Specifically, the set of natural numbers (1, 2, 3, 4, 5, and so on).
Let’s sample one of his arguments, from later down in the same post:
First, we need to define our terms. “Infinite” or “infinity” has many definitions, some better than others. I will focus on two definitions: the standard one, and then a superior one.
The standard definition of “infinite” means “never-ending”, “limitless”, or “without boundaries.”
The superior definition of “infinite” means “without inherent limitation.” These two definitions often get mushed together, and it results in conceptual confusion.
The difference between these two definitions is metaphysical, as I will explain. Take the question:
“How many positive integers are there?”
The standard response is, “There is an infinite amount” – implying that there is an “actually-infinite” amount. That somehow, you can put “all the positive integers” into a set, and the amount of elements you’ll end up with is “infinity”.
In fact, mathematicians have a term for the actual size of the set of positive integers. They call it “Aleph-null.” According to modern set theory, originally conceived by Georg Cantor, Aleph-null is the smallest size of infinity. Mathematicians think there are different actual sizes of infinite sets.
This is nonsense and a confusion about the metaphysical status of numbers, which I’ll get into later. A superior response to the question, “How many positive integers are there?” is to say:
“There is no inherent limitation to the size of set you can create with positive integers.”
That doesn’t mean there’s an actually-infinite set out there in the world. It means there’s no limit to the size of the set’s construction.
Now, there are a few holes in this reasoning (hang on, we are going somewhere good!) that tie back to a key question, what is Mathematics? Not, what does the word mean but: what is the substance, the essence of the discipline and what it studies.
The best answer I have found, building on what was said by a distinguished professor long ago now, in an aside in my Uni’s good old N2 course, M100: [the study of] the logic of structure and quantity.
What is being missed here, is that there is a structure being laid down by going, 0,1,2 . . . k, k+1,k+2 . . . n, n+1 . . . and that it has an associated, countable scale or quantity. one, that we can label aleph null then study as a quantity in its own right. Related, we can use ordinality as a structure to develop ordinal and even transfinite numbers.
Where, we can also conceive of an extension to the number line with a value H beyond any n in the natural counting numbers, N. Then, use our favourite catapult 1/x to associate h = 1/H, number closer to 0 than 1/n for any number n we can actually complete counting to. H is a transfinite hyperreal, h is a tamed infinitesimal as was envisioned by Archimedes and co., later by Newton, Leibniz and Euler et al. It took Robinson to tame such. The hyperreals are set R*.
(NB: Notice, I am here further defining n, where n+1 is obviously also finite and BOUNDS n by succeeding it. That is how n is finite, it is a whole number value that mileposts the reals line R and is bounded by onward integers. The infinite, by contrast is any quantity that cannot be bounded by at least one definite, finite value. That is the proper sense of infinite as meaning beyond bounds or limitless. The limit in question, being itself finite. There is thus no conceptual barrier here to greater and greater transfinite numbers, ordinal or cardinal. Also, the catapulting between h and H via 1/x unifies the extended number line R*. The basic number line quantities are a unified whole and can be assigned to definite collections, i.e. sets. Namely, N,Z,Q,R,R*. Notice, I have skipped C, which is a two dimensional vector domain created from the reals and a second axis rotated by 90 degrees, the so-called imaginary numbers, in fact vectors of rotation.)
In short, there is room for transfinite numbers of scale — notice, scale, order of magnitude — aleph-null and beyond.
Back to Patterson:
Mathematicians use phrases like:
“The set of all positive even integers.”
They claim the size of that set is infinite – specifically, it is “Aleph-null”, which is the smallest infinity. Infinite sets with larger cardinalities are called “Aleph-one”, “Aleph-two”, and so on. There are, according to mathematicians, an infinite amount of sizes of infinite sets. This was the ground-breaking work of Georg Cantor, on top of which modern mathematics is built.
Now, instead of referencing “the set of all positive even integers”, imagine we’re talking about “the set of all positive odd integers.”
The cardinality, as you might intuitively think, is the same. Aleph-null.
What about the question:
“What is the cardinality of the set of all even and odd integers together?” In other words, what is Aleph-null plus Aleph-null?
The answer: Aleph-null. The cardinalities are the same.
If this strikes you as logically contradictory, that’s because it is, but mathematicians have believed this for over a century.
This means they accept the following idea: a whole can be the same size as its constituent parts, because “Aleph-null” is the same size as “Aleph-null plus Aleph-null.”
Nope, and it is not because there is an elementary error:
They justify this by saying, “Regular finite logic doesn’t apply when talking about infinite things!”
The real issue is of course the implications of there being no finite bound:
0, 1, 2, 3, 4, 5, 6 . . .
0, 2,4,6 . . . = 2×0, 2×1, 2×2, 2×3, . . . 2xn, . . .
1,3,5 . . . = 2×0 +1, 2×1 +1, 2×2+1 . . . 2xn+1 . . .
k, k+1,k+2 . . . = k +0, k+1, . . . –> k-k+0, k+1-k, k+2 -k . . .
That is, once there is no finite bound involved, N can be transformed into a great many sets that have the same scale, a quantity that can be labelled aleph null and specified as the size of N. That is, a scale such that a set can be without limit put into 1:1 correspondence with N. Logic of structure and quantity at work again, which we duly need to study.
So, the Mathematicians are quite correct, once there is not a finite bound, sets similar to the above, though seemingly smaller than N have the same scale as N. Strange, but not incoherent. We just need to accept a paradigm shift.
As usual.
This becomes even more interesting as Patterson unveils his underlying concepts:
In order to understand the refutation of Cantor’s Diagonal Argument, we have to understand the metaphysics of mathematics – what numbers are, and their relationship to our minds.
In a nutshell: numbers are concepts. They do not exist separate from our minds, nor do they exist separate of our conception of them.
Numbers (15, 2501, 56, etc.) are symbols used to represent concepts – concepts dealing with amount, magnitude, and quantity. Those numbers are just like letters and words. When we construct a sentence out of letters, we’re arranging some visual medium in such a way that evokes concepts in the minds of the reader.
The same is true in mathematics. The symbols of “+” and “-“ do not reference objective entities in the world. They are simply shorthand – a visual symbol – for a logical relation between our concepts.
See the key contrast? Namely, “numbers are concepts. They do not exist separate from our minds, nor do they exist separate of our conception of them.” and again, “The symbols of “+” and “-“ do not reference objective entities in the world.”
That is, Objective is here used to denote tangible and external to mind, what is not like that is thus deemed not objective, it is subjective; a human invention. The abstract, in his thought, is inherently subjective, don’t even mention Plato’s silly world of forms. This framing, however, is an error. One, reflective of the baneful effects of scientism and relativism.
Instead, start afresh from a basic observation: we are finite, fallible [= error prone], morally struggling, too often ill-willed and even stubborn. So, our first person perceptions, awareness, sense of location and orientation in the world, beliefs, opinions, reasoning claims, knowledge claims etc fall under this concern. So, we need warranting filters that can improve the reliability of such experiences, without falling into hyperskepticism. Which, more effectively defines objectivity:
In philosophy, objectivity is the concept of truth independent from individual subjectivity (bias caused by one’s perception, emotions, or imagination). A proposition is considered to have objective truth when its truth conditions are met without bias caused by a sentient subject.
Yeah, that’s Wikipedia testifying against known interest. We may as well acknowledge when they get something that may easily have been ideologically loaded right.
Hence, too, the centrality of warranting filters in establishing objectivity. The objective is generally knowable because it has been adequately filtered from the error proneness involved in our first person experience subjectivity.
Notice, truth is accurate description of entities or states of affairs etc. Nothing in that, requires that we have concrete, tangible external objects such as a coconut tree. Abstracta such as numbers can be objective and manifest their presence in something as simple as clustering fingers into a three and a two then joining them as a five, illustrating how || + ||| –> |||||. That is an inherent pattern that holds in any possible world. Even, the seemingly silly world of forms has a germ of truth in it, abstracta eternally contemplated by an utterly wise necessary being; building on a point from Augustine.
Where, many abstracta can be expressed in words or other symbols and so are communicable. Warrant regarding abstracta at the core of Mathematics is like that. For example, consider von Neumann’s construction:
{} –> 0
{0} –> 1
{0,1} –> 2
. . .
{0,1.2 . . .} –> w, omega, first transfinite ordinal.
Of course, w is not a finite bound.
So, when we see from Patterson,
There is no “largest possible [natural, counting] number.” That’s not how numbers work. Any number N that you conceive of, I can always think of N+1. Does that mean that N+1 exists prior to its conception? Certainly not.
See how the underlying radically relativist constructiv-ISM was brought in? Long since, w as a definable ordinal number was brought in, and bounds any n in N that can be exceeded by an equally finite n+1. So, there is no definable last finite, f so f+1 = w. That is we have a fuzzy border zone for N, but we can identify what it takes to be a member and what would not be a member, N is a valid though transfinite set. And all of this is objective.
We therefore see that objectivity is a pivotal quantity and how warranting filters help us achieve it. This is not to denigrate our first person experience, but it allows us to address a key limitation, error proneness.
Bonus, we have a clearer vision of the transfinite and of Mathematics. END
L&FP, 49: Debating the validity (and objectivity) of infinity
Except infinity does not exist so how can it be objective?
Here we go again.
From the article: “That doesn’t mean there’s an actually-infinite set out there in the world”
True, but that is different than the concept of infinity being mathematically valid.
In our discussions, we have made the distinction between potential and actual infinities. This guy is emphasizing potential infinities, but I think he is wrong to dismiss actual infinities as valid mathematics.
I just read up on this guy Patterson. He seems like a crank. Here’s from his about page,
Not sure why KF bothered to highlight him?
VL, If you read the OP, I used him as an example of error, and showed how pervasive errors of relativism and subjectivism are. Of course, he erred on the abstract, objective reality of numbers, especially transfinites. That’s common. HOW he erred is an example to learn from, where for example he failed to realise how various subsets of N can be transformed into N giving 1:1 correspondence and of course drawing out the implication of no finite bound. KF
Jerry, what do you mean by “infinity” and its non existence? Infinity is a label for a particular property, that which has no finite bound. It is for example a property of several key sets of numbers, sets that are transfinite, indeed, we can identify particular numbers that are transfinite, w and family come to mind. The number of points in a continuum will be finite, line, area, volume etc. Over in theology, it is used to describe how God has no finite bound to his characteristics, though there is a logical requisite of compossibility and there is the moral limit of utter goodness. We are part of an evidently finite physical cosmos and it is finitely bound in the past due to the supertask of traversing the transfinite in finite stage steps. There is nothing incoherent in transfiniteness or more broadly infinity, by contrast with a seven sided pentagon or the notion that God is challenged to make a rock so massive he cannot move it . . . a classic example of forms of words that are meaningless. KF
Does not exist in real world.
Cannot point to anything infinite.
Only exists in our imagination. We have been through this a few times before.
If infinity exists, then so do weightless, pink elephants with yellow polka dots ridden by Obi Wan Kenobi on a frictionless surface.
Is that objective?
If it is, then you’re in Murray’s world now.
Aside: the concept of infinity is extremely useful as is most mathematics that uses it. But that does not mean it exists.
Sabine’s take
https://www.youtube.com/watch?v=Bq9xR5PUs6s
–Ram
Jerry, real world is a slippery term, I imagine you mean that there are no infinite values for concrete, tangible entities. Or for span of the past or the like. But just because something or some state of affairs is abstract does not mean it is non objective, is a matter of our imagination, it does not express actual realities. KF
So imagination is objective.
This means that anything in my imagination whether it is possible in any world is objective. Makes the word “objective” sort of useless.
Thus, the tooth fairy is objective or any other wild thing I can dream of objective.
jerry,
What it means is that you’re living in your own head. It’s delusion to some degree. For various reasons you believe what you believe. Not everyone you meet will agree with what’s in your own head. I.e, your semi-delusion. But you have my permission to enjoy your delusions for as long as they lasts.
Steal this book
–Ram
Jerry, that one imagines is self evident to oneself. That one may communicate regarding same among a race of similarly imaginative agents would mean that beyond reasonable doubt the fact of our being imaginative is common knowledge, e.g. I just imagined a coconut tree. That fact of our being an imaginative race is objective and generally known. If one is a credible habitually truthful person, one’s report about what s/he imagined may have enough credibility to be reasonably warranted as common sense truth. For example, people who have had traumatic experience commonly report flashbacks. I just had one on a funeral of someone near and dear to me, triggered by simply thinking on related matters. Whether the content of imagination or broader contemplation is accurate to say mathematical or empirical reality of tangible objects is another matter; one, for warranting filters. That is how the thoughts in the OP on transfinite numbers and magnitudes take up objectivity. Which, informed by the criterion of adequate warrant in the face of error proneness, is anything but what a now over-used dismissive term — meaningless — suggests. I beg to remind one and all that the verification principle beloved of positivists fell to self referential incoherence something like sixty years ago. KF
Ram, Hossenfelder is quite good. KF
What is in your mind or imagination may or may not have a high correlation with the real world.
That there is a Walmart 5 miles from my house has an extremely high correlation with my images of it. All my neighbors and myself have no problem finding it and shopping there
There is no such correlation of infinity with the real world. No one can find an example of it in the real world. So just as the weightless elephant is absurd, infinity is absurd. However, this does not mean that certain ideas that have no correlation with the real world are not useful. Usefulness, however does make them any more objective than any other imagination anyone has.
We have had these discussions before.
If you want to say we all imagine, then that is obvious. Is anyone doubting that? I doubt they are except for one deranged person here who actually does not believe it and is just playing games.
Jerry, real world is not simply equal to concrete tangible entities. For, there are many abstracta that are very real. A relationship of being a cousin, or a woman [membership of a class], or having love or being loved, or twoness, or evenness or being a transfinite quantity or number would all fit in, and many of these are tied to the substructure of this or any other possible world. Admittedly, the countable transfinite is not to be found instantiated in the empirical world but it is directly implied by the structure of a phenomenon we observe effects of, countable number. KF
Sounds like another commenter here. Have you caught the same disease?
You have conflated the external physical world with our imaginations and lumped them together and called them the real world.
I don’t want to get into a never ending discussion of each perception we have. It will lead no where that is productive. But just as an example we use the mathematical concept of infinity to help solve issues in the physical world external to us. In that way they are extremely useful even if infinity does not exist in the real world but only in imaginary ones.
Before you repeat that our imagination is the real world, then so is weightless pink unicorns with yellow polka dots and ridden by the avatar of our choice. We can even draw them in the physical world.
You are in a hole. Get out of it instead of digging it deeper.
Jerry, kindly note the significance of mathematical entities. Also the role of warrant in objectivity, what is imaginable or imagined may or may not be actual or possible. Some few things cannot even be imagined such as a seven sided pentagon. Again, reality is not to be conflated with tangible and concrete. KF
Jerry: There is no such correlation of infinity with the real world. No one can find an example of it in the real world. So just as the weightless elephant is absurd, infinity is absurd. However, this does not mean that certain ideas that have no correlation with the real world are not useful. Usefulness, however does make them any more objective than any other imagination anyone has.
I don’t understand why you keep waving this flag. Okay, so there are no physical infinite sets. So what? You admit the concept, in the abstract, may or is still useful so . . .
I don’t understand what this whole thread is for. Except to make fun of Steve Patterson. I guess. He just seems to want some attention and, guess what, we gave It to him. Viola Lee even checked his blog.
Move on, nothing of interest here.
JVL, the thread from OP shows a needed corrective and clarification. Donkey work, but useful and needed, this is basics. Over in another thread we have struggles over self origination. KF
PS, bonus, there is clarification of the difference between concrete tangibility and objective reality.
The concept of truth has no physical characteristics and cannot be figured a concept that is “more perfect” or that is beyond the truth. Everyone knows what is the truth as a concept that lives in the mind , everyone is in the hurry and happy to tell you the truth about something . Truth is perceived more real and more objective than the existence of an apple .The truth is also infinite unless you find a number that is bigger than truth. 😉
PS: All the meta-values are immaterial, objective and infinite.
Objective reality – Pink weightless elephants with yellow polka dots and three horns that can swim the breaststroke.
Concrete tangibility – The Walmart off of Rt 1 in Seabrook, New Hampshire.
Aristotle is glad to see the update in understanding and terminology.
Jerry,
your idea as you report has objectivity as regards having an idea, that idea refers to imagined concrete tangible entities that embed a contradiction of massless hadron based entities. Such will not exist, due to contradiction of core characteristics, i.e. massless hadron based elephants — impossible of being, I suspect also that elephants cannot swim the breaststroke.
There have been elephants in the loose sense with multiple tusks.
I use the stricter term as there is a definition of weightlessness that can apply to a massive entity: the force a body in a gravity field exerts on anything that freely supports it.
The Walmart can and credibly does exist.
Such have nothing to do with say the null set, von Neumann’s extension into N, thence N,Z,Q,R,R* and objectivity — tracing to warrant — for same, as well as pervasiveness in phenomena of structure and quantity tied to such. The unlimited extension of counting numbers is readily shown and that is a definable quantity without finite bound. It comes with the counting numbers. Continuum is a logical extension of continuity in space, and that is another demonstrably larger transfinite.
The power set series from aleph null is warranted and the ordinals from w also. With more.
Abstract warranted entities like that have objectivity similar to other abstract entities, attributes and states of affairs as noted, being a woman, loving, etc. Further to such, information and its measurable quantity are abstract and objective.
I would argue that how we are forced to use operational definitions of energy indicates its abstract nature. Time is riddled with abstracta, thus energy-time relationships studied in thermodynamics, which of course are riddled with mathematical abstracta. Then we can point to geometry and its system of properties and relationships, which IIRC was viewed by Plato as an ideal case of the objectively true and real; I stand to be corrected but I doubt Aristotle denied objectivity though he disputed the notion of a world of forms, preferring a concept of abstraction from concrete cases. And more.
Objective reality and concrete tangible materiality are not equivalents.
Down that road lie the errors of materialism and broader physicalism.
KF
F/N: Springer Enc of Math on Finitism:
https://encyclopediaofmath.org/wiki/Finitism
Wiki gives background:
I cannot but note a contrast of attitude to Wiki’s infamously libellous anti-ID fulminations!
I found this too:
https://www.extremefinitism.com/
What becomes significant is the physicalism involved, here quite explicitly though probably unconsciously. Such, of course, is self refuting, undermining mind and its credibility. Computation on a GIGO constrained substrate is not reason of a free, self-moved agent.
It should also be clear that Godel struck a fatal blow to Hilbert’s programme. The infinite is here to stay and there is no good reason to refuse objective reality to the null set and its consequences such as N,Z,Q,R,R* etc. For C, I think the rotating vectors view is decisive.
More can be said, this is backdrop.
KF
F/N: I find it interesting to follow up our latest proponent of finitism, Karma Peny (who is by his admission a “staunch atheist”):
https://www.extremefinitism.com/blog/what-is-a-number/
Now of course, generally Mathematicians routinely use transfinites and have moved on since Godel. Mathematics, often being grandfathered in as a “Science,” despite its blatantly non-empirical character. But our interest is in how scientism, materialism and the like are influencing thinking here.
We see, first a discussion on specific values that have been represented. This side steps the von Neumann type construction (see OP) which does show how each number from 0 has a successor by increment. So for every identified or symbolised n in N, there is n+1 also in the set, and 0 is in the set, by definition. We have a trivial proof by induction on N, no need to go to elaborate transfinite induction, the onward succession of order types is such that once one accepts w as order type of N as a whole, succession continues, w+1 etc.
Let me clip and augment:
So, confining numbers to what we can state as naturals (and presumably Reals or Complex values) is not a valid criterion of being a relevant quantity to be termed a number. Once we may reasonably define and represent a quantity in a structure of quantities and reason coherently regarding such, we have valid numbers.
The confining to brains as computational substrates is interesting. Surely, the human mind is responsibly and rationally free and we can define symbols to represent naturals etc, also infinitesimals and transfinites, applying suitable rules and logic. That is enough.
Of course, we are finite and fallible, but we have succeeded in developing the relevant Mathematics.
Our source continues:
See the imposed physicalism and associated computationalism? Ironically, the breakthrough in Mathematical power created by algebra and symbolisation is missed here. Algebras, allow us to move to the next level.
Of course the symbols point to abstracta, which are implicitly dismissed and derided by setting up Plato as strawman. But, we need only apply the point that abstracta are implicit in many cases, can be represented and reasoned about effectively. There is nothing suspect about n+1 in N, or w and w+1 in the transfinite hyperreals.
He goes on:
Ahem, little more than error carried forward.
Setting up a strawman.
Ahem, the endlessly is rather the point is it not? And per von Neumann, the logic of the transfinite emerges.
Now, I am not just picking on cranks out there, I am pointing to patterns of thinking and to how we have lost the core, recognising that there are responsible objective bodies of knowledge pivoting on warrant that needs to be taken seriously and addressed on the merits if we are even in disagreement. Yes, even that minor paradigm called design theory, Wiki moderators.
There is importance in recognising that Mathematics transcends the physical domain and is objective.
KF
PS, is physicality equivalent to reality or existence?
F/N: At a more serious level, we may briefly ponder intuitionism and linked constructivism in Mathematics. SEP:
Constructivism then goes, per IEP:
These, of course have radical consequences.
However, there seems to be a substitution of the epistemology of strong warrant [proof acceptable to the community of practitioners of Math) for the ontology of what is the case even if we do not yet or even may never demonstrate to be so. This is why an implication would be distinguished from its Boolean Algebra equivalents. However, there seems to be no really good reason for such a substitution, we for cause distinguish warranted knowledge from accurate description of states of affairs.
From which, we then can see that there is no good or compelling reason to reject a proof based on A XOR ~A, But ~A is absurd so A which pivots on excluded middle. Yes, it may be desirable to chain from axioms and a body of interconnected established results to a new one, step by logical step but that does not compel us to dismiss reductio and associated principles.
In short, I here deny that we CONSTRUCT or INVENT thje substance of mathematics which then comes into existence by that construction. Instead, in part, we build up our body of knowledge by relevant derivations. Knowledge recognises truth or credible truth on warrant, it does not create truth that did not exist prior to that point.
KF