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L&FP, 49: Debating the validity (and objectivity) of infinity

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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

26 Replies to “L&FP, 49: Debating the validity (and objectivity) of infinity

  1. 1
    kairosfocus says:

    L&FP, 49: Debating the validity (and objectivity) of infinity

  2. 2
    jerry says:

    Except infinity does not exist so how can it be objective?

    Here we go again.

  3. 3
    Viola Lee says:

    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.

  4. 4
    Viola Lee says:

    I just read up on this guy Patterson. He seems like a crank. Here’s from his about page,

    In my own research, I have discovered a remarkably consistent truth: orthodox opinions are almost always wrong. The “mainstream consensus” on any given topic – whether about political theory, quantum physics, or the use of infinity in mathematics – frequently makes foundational errors. Thus, my worldview looks radical when compared to the mainstream, and it also looks like crankery, since I make outrageous claims like having a resolution to the mind-body problem.

    Not sure why KF bothered to highlight him?

  5. 5
    kairosfocus says:

    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

  6. 6
    kairosfocus says:

    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

  7. 7
    jerry says:

    what do you mean by “infinity” and its non existence?

    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.

  8. 8
  9. 9
    kairosfocus says:

    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

  10. 10
    jerry says:

    just because something or some state of affairs is abstract does not mean it is non objective

    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.

  11. 11
    ram says:

    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

  12. 12
    kairosfocus says:

    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

  13. 13
    kairosfocus says:

    Ram, Hossenfelder is quite good. KF

  14. 14
    jerry says:

    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.

  15. 15
    kairosfocus says:

    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

  16. 16
    jerry says:

    real world is not simply equal to concrete tangible entities

    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.

  17. 17
    kairosfocus says:

    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

  18. 18
    JVL says:

    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.

  19. 19
    kairosfocus says:

    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.

  20. 20
    Lieutenant Commander Data says:

    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.

  21. 21
    jerry says:

    there is clarification of the difference between concrete tangibility and objective reality

    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.

  22. 22
    kairosfocus says:

    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

  23. 23
    kairosfocus says:

    F/N: Springer Enc of Math on Finitism:

    https://encyclopediaofmath.org/wiki/Finitism

    Finitism

    A methodological point of view, due to D. Hilbert, as to what objects and methods of argument in mathematics should be counted as absolutely reliable. The main requirements of finitism are:

    1) the objects of arguments are constructive objects (cf. Constructive object), for example the written form of natural numbers, formulas in symbolic language, and finite collections of them;

    2) the operations that can be applied are uniquely defined and can in principle be performed (are computable);

    3) one never considers the set of all objects x
    of any infinite collection; a general judgment A(x) is a statement about an arbitrary object x

    that one can confirm in each particular case;

    4) the assertion that there exists an object x
    with the property A(x)

    means that one can either produce a concrete example of such an object or show a way of constructing one.

    The restriction of finitism on logic is close to intuitionism, but on the whole the finitary point of view is more rigid. An argument that satisfies the requirements 1)–4) does not go beyond the bounds of intuitionistic arithmetic (see Intuitionism).

    After being formalized (see Axiomatic method), substantial mathematical theories become constructive objects (collections of constructive objects). Within the bounds of the approach of Hilbert and his followers, finitism is necessary for studying such formalized theories; only those properties of theories that can be proved by finitary methods are counted as reliable. The Gödel incompleteness theorem showed that finitary methods are insufficient as foundations of mathematics. This led to the need to extend the methods that can be applied in proof theory beyond the bounds of finitism.

    Wiki gives background:

    The introduction of infinite mathematical objects occurred a few centuries ago when the use of infinite objects was already a controversial topic among mathematicians. The issue entered a new phase when Georg Cantor in 1874 introduced what is now called naive set theory and used it as a base for his work on transfinite numbers. When paradoxes such as Russell’s paradox, Berry’s paradox and the Burali-Forti paradox were discovered in Cantor’s naive set theory, the issue became a heated topic among mathematicians.

    There were various positions taken by mathematicians. All agreed about finite mathematical objects such as natural numbers. However there were disagreements regarding infinite mathematical objects. One position was the intuitionistic mathematics that was advocated by L. E. J. Brouwer, which rejected the existence of infinite objects until they are constructed.

    Another position was endorsed by David Hilbert: finite mathematical objects are concrete objects, infinite mathematical objects are ideal objects, and accepting ideal mathematical objects does not cause a problem regarding finite mathematical objects. More formally, Hilbert believed that it is possible to show that any theorem about finite mathematical objects that can be obtained using ideal infinite objects can be also obtained without them. Therefore allowing infinite mathematical objects would not cause a problem regarding finite objects. This led to Hilbert’s program of proving both consistency and completeness of set theory using finitistic means as this would imply that adding ideal mathematical objects is conservative over the finitistic part. Hilbert’s views are also associated with the formalist philosophy of mathematics. Hilbert’s goal of proving the consistency and completeness of set theory or even arithmetic through finitistic means turned out to be an impossible task due to Kurt Gödel’s incompleteness theorems. However, Harvey Friedman’s grand conjecture would imply that most mathematical results are provable using finitistic means.

    Hilbert did not give a rigorous explanation of what he considered finitistic and referred to as elementary. However, based on his work with Paul Bernays some experts such as William Tait have argued that the primitive recursive arithmetic can be considered an upper bound on what Hilbert considered finitistic mathematics.

    As a result of Gödel’s theorems, as it became clear that there is no hope of proving both the consistency and completeness of mathematics, and with the development of seemingly consistent axiomatic set theories such as Zermelo–Fraenkel set theory, most modern mathematicians do not focus on this topic. Today, most mathematicians are considered Platonist and readily use infinite mathematical objects and a set-theoretical universe . . . .

    In her book The Philosophy of Set Theory, Mary Tiles characterized those who allow potentially infinite objects as classical finitists, and those who do not allow potentially infinite objects as strict finitists: for example, a classical finitist would allow statements such as “every natural number has a successor” and would accept the meaningfulness of infinite series in the sense of limits of finite partial sums, while a strict finitist would not. Historically, the written history of mathematics was thus classically finitist until Cantor created the hierarchy of transfinite cardinals at the end of the 19th century . . . .

    Leopold Kronecker remained a strident opponent to Cantor’s set theory:[1]

    God created the integers; all else is the work of man.[2]

    Reuben Goodstein was another proponent of finitism. Some of his work involved building up to analysis from finitist foundations.

    Although he denied it, much of Ludwig Wittgenstein’s writing on mathematics has a strong affinity with finitism.[3]

    If finitists are contrasted with transfinitists (proponents of e.g. Georg Cantor’s hierarchy of infinities), then also Aristotle may be characterized as a finitist. Aristotle especially promoted the potential infinity as a middle option between strict finitism and actual infinity (the latter being an actualization of something never-ending in nature, in contrast with the Cantorist actual infinity consisting of the transfinite cardinal and ordinal numbers, which have nothing to do with the things in nature):

    But on the other hand to suppose that the infinite does not exist in any way leads obviously to many impossible consequences: there will be a beginning and end of time, a magnitude will not be divisible into magnitudes, number will not be infinite. If, then, in view of the above considerations, neither alternative seems possible, an arbiter must be called in. —?Aristotle, Physics, Book 3, Chapter 6 . . . .

    Ultrafinitism (also known as ultraintuitionism) has an even more conservative attitude towards mathematical objects than finitism, and has objections to the existence of finite mathematical objects when they are too large.

    Towards the end of the 20th century John Penn Mayberry developed a system of finitary mathematics which he called “Euclidean Arithmetic”. The most striking tenet of his system is a complete and rigorous rejection of the special foundational status normally accorded to iterative processes, including in particular the construction of the natural numbers by the iteration “+1”. Consequently Mayberry is in sharp dissent from those who would seek to equate finitary mathematics with Peano Arithmetic or any of its fragments such as primitive recursive arithmetic.

    I cannot but note a contrast of attitude to Wiki’s infamously libellous anti-ID fulminations!

    I found this too:

    https://www.extremefinitism.com/

    Finitism is a philosophy of mathematics that accepts the existence only of finite mathematical objects.

    Strict Finitism is a form of finitism that rejects objects known as ‘countably infinite’.

    Extreme Finitism goes much further as it rejects all uses of the concept of ‘infinity’.

    Extreme Finitism is the philosophy that only accepts the existence of finite objects and where some symbols are allowed to represent process definitions where no end point has been specified. Furthermore, Extreme Finitism opposes the use of words that imply that ‘infinity’ is a valid concept.

    These words include ‘infinitely many’, ‘an infinite number of’, ‘infinite’, ‘to infinity’, ‘continuous’, ‘analogue’, ‘forever’, ‘eternally’, ‘infinitesimal’ and all other words and phrases that imply ‘infinity’.

    Extreme Finitism views a quantity as merely a property of something that does/could exist. A quantity is not an object in itself and it cannot be said to have its own existence. Suppose someone picks up two apples and discovers one has a rough texture and the other has a smooth texture. The textures have no inherent existence of their own; they are merely properties of other things that exist, namely the apples. We perform an examination/measurement process on the apples to determine the type of texture.

    Similarly we could perform an examination of how many apples someone is holding, and our examination process might return a value of 2. Here we have a defined group that is ‘things the person is holding’, and our number 2 is a count returned from the examination/measurement process that ‘counts apples’ for the defined group. If the person does not exist, then we cannot perform the examination/measurement and so the corresponding numeric property cannot be evaluated. So, in the same way that texture is a property of something that exists, what we call a ‘natural number’ is a property of a defined group that exists.

    Fundamental to modern mathematics is the claim that numbers have their own existence, and that mathematics is detached from physical reality. If we accept these claims, it can appear logical to say that if a natural number (n) exists then its successor (n+1) must also exist, thus the amount of natural numbers cannot be finite. We appear to have used logic to deduce that infinitely many of something is a valid concept. The basis of this dubious logic is that things like natural numbers can be said to ‘exist’ and that such objects are somehow detached from all physical reality.

    A number appears to have a physical presence when we write one down (it physically exists on the paper), or when we store one on a computer (it physically exists in the state of voltages), or even when we think about one (it physically exists in the state of our brain chemistry). Therefore it could be argued that a number stored on a computer, say, has its own physical presence and as such has its own existence. A physical memory location might hold a value corresponding to a count of 2, but arguably this is just a description of a generic property. Similarly we could store the text “rough texture” on the computer, but this does not prove that a rough texture can have its own existence independent of all physical reality.

    Perhaps our starting assumption should be that the only thing that exists is a shared physical reality that we refer to as the real world. Anything we imagine is just the result of brain chemistry, and just like computer memory, brain chemistry is a finite physical entity. Neither a biological brain nor an electronic computer can imagine (or visualise or picture) infinitely many of something, but perhaps they can delude themselves into accepting that they can? Perhaps we should not accept that we can imagine or think-of or otherwise conceive of anything that has no relation to the real world, perhaps we can only imagine that we can (in other words, perhaps the belief that we can conceive of non real world things is self delusion)? Perhaps we should re-invent mathematics from scratch, where every concept in our new mathematics has a basis in reality. Then we would never have to ask children to believe they can imagine what the square root of minus 1 is. Instead we might ask them to visualise a grid with the four directions ‘+’ (right), ‘-‘ (left), ‘+i’ (up), ‘-i’ (down) and a movement-based concept of multiplication so that they can easily visualise the movement sequences (corresponding to i squared and minus i squared) that reach the ‘one-square-in-the-left-direction’ position. This re-invention of mathematics should contain no paradoxes, no divide by zero issues, and no claims that we can work with actual infinities. In a new mathematics we could have a geometry based on a tangible definition of a point instead of a definition that says a point has no parts to it (currently a point is effectively defined as being nothing!). A natural number could be a measurable quantity instead of an abstract concept that has no connection with reality. Perhaps we should not even use the word ‘mathematics’ because maybe the fundamentals are really all about computing?

    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

  24. 24
    kairosfocus says:

    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/

    Consider the statement: If n exists then n+1 exists

    Now if ‘n’ in this statement refers to a written numeral, then this statement is probably false because as far as we know only a finite amount of numerals have ever been written. Also, if brains are mere finite biological machines of some sort, then to conceive of a number might be comparable to a computer process accessing a symbol in a memory location. And if numbers are merely physical symbols at various locations inside brains, then just like written numerals, it is likely that only a finite amount of them can ever exist.

    But we like to believe that natural numbers are not physical in any way. We might say that for any given natural number (n) its successor (n+1) must exist and therefore infinitely many natural numbers must exist. This is a cyclic argument because we cannot show that numbers have their own out-of-brain existence. We must first assume that infinitely many non physical numbers exist before we can make this argument to show why infinitely many numbers must exist. This belief in the existence of non physical or ‘abstract’ objects is called ‘Platonism’ after the Ancient Greek philosopher Plato.

    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:

    {} –> 0
    {0} –> 1
    {0,1} –> 2
    . . .
    expand:
    {0,1,2 . . .n} –> n+1
    {0,1,2 . . . n+1} –> n+2
    continue, without limit, i.e. we point to continuation
    [This is a “finitary” step, of using finite steps of
    reasoning to lead to a conclusion that is of infinite character]
    . . .
    {0,1.2 . . .} –> w, omega, first transfinite ordinal.

    Of course, w is not a finite bound.

    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:

    Many modern mathematicians try to distance themselves from Platonism by calling themselves ‘formalists’. They claim to reject the notion that numbers are out there somewhere, mysteriously existing in a non physical form.

    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:

    Instead of claiming that all numbers ‘exist’, they [the Formalists] say that mathematics is just a game we play with symbols and rules. For example, with just ten symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and a finite number of rules (such as Rule_1: Each symbol by itself is a number; Rule_2: If you append one or more symbols to a number, where the left-most symbol is not allowed to be zero, you get another number) then this allows for an infinite number of expressions.

    The claim is that these symbols and rules are not subject to physical limitations. Yes we can only use them to construct a finite quantity of numbers, but this doesn’t matter. Merely by appreciating what can be done with these ten symbols and these two rules we can say that we have conceived of something that ‘defines an infinite number of expressions’. Bizarrely the claim is that we can conceive of the infinite by describing how to do something and saying it can be done repeatedly.

    Ahem, little more than error carried forward.

    But again, if we relate these rules to written numerals then the argument fails. The rules merely provide instructions on how we might construct other numbers, they do not tell us how we can achieve a quantity of them that is not finite.

    Setting up a strawman.

    Even if we were to add a third rule that says to keep constructing different numbers endlessly then all it would be saying is to keep increasing a finite quantity to a bigger but still finite quantity.

    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

  25. 25
    kairosfocus says:

    PS, is physicality equivalent to reality or existence?

  26. 26
    kairosfocus says:

    F/N: At a more serious level, we may briefly ponder intuitionism and linked constructivism in Mathematics. SEP:

    Intuitionism is a philosophy of mathematics that was introduced by the Dutch mathematician L.E.J. Brouwer (1881–1966). Intuitionism is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true, and the communication between mathematicians only serves as a means to create the same mental process in different minds.

    This view on mathematics has far reaching implications for the daily practice of mathematics, one of its consequences being that the principle of the excluded middle, (A OR ~A), is no longer valid. Indeed, there are propositions, like the Riemann hypothesis, for which there exists currently neither a proof of the statement nor of its negation. Since knowing the negation of a statement in intuitionism means that one can prove that the statement is not true, this implies that both A and ¬A do not hold intuitionistically, at least not at this moment. The dependence of intuitionism on time is essential: statements can become provable in the course of time and therefore might become intuitionistically valid while not having been so before.

    Constructivism then goes, per IEP:

    Constructive mathematics is positively characterized by the requirement that proof be algorithmic. Loosely speaking, this means that when a (mathematical) object is asserted to exist, an explicit example is given: a constructive existence proof demonstrates the existence of a mathematical object by outlining a method of finding (“constructing”) such an object. The emphasis in constructive theory is placed on hands-on provability, instead of on an abstract notion of truth. The classical concept of validity is starkly contrasted with the constructive notion of proof. An implication (A=>B) is not equivalent to a disjunction (~A OR B), and neither are equivalent to a negated conjunction (~(A OR ~B)). In practice, constructive mathematics may be viewed as mathematics done using intuitionistic logic.

    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

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