A Finitist's Manifesto: Do we need to Reformulate the Foundations of Mathematics?
aa r X i v : . [ m a t h . L O ] S e p A Finitist’s Manifesto: Do we need to Reformulatethe Foundations of Mathematics?
Jonathan Lenchner
There is a problem with the foundations of classical mathematics, and potentially even with thefoundations of computer science, that mathematicians have by-and-large ignored. This essay is acall for practicing mathematicians who have been sleep-walking in their infinitary mathematicalparadise to take heed. Much of mathematics relies upon either (i) the “existence” of objects thatcontain an infinite number of elements, (ii) our ability, “in theory”, to compute with an arbitrarylevel of precision, or (iii) our ability, “in theory”, to compute for an arbitrarily large number of timesteps. All of calculus relies on the notion of a limit. The monumental results of real and complexanalysis rely on a seamless notion of the “continuum” of real numbers, which extends in the planeto the complex numbers and gives us, among other things, “rigorous” definitions of continuity, thederivative, various different integrals, as well as the fundamental theorems of calculus and of algebra– the former of which says that the derivative and integral can be viewed as inverse operations, andthe latter of which says that every polynomial over C has a complex root. This essay is an inquiryinto whether there is any way to assign meaning to the notions of “existence” and “in theory” in(i) to (iii) above.On the one hand, we know from quantum mechanics that making arbitrarily precise measure-ments of objects is impossible. By the Heisenberg Uncertainty Principle the moment we pin downan object, typically an elementary particle, in space, thereby bringing its velocity, and hence mo-mentum, down to near 0, there is a limit to how precisely we can measure its spatial coordinates.In symbols: σ x σ p ≥ h π , where σ x is the standard deviation of position, σ p is the standard deviation of momentum, and h isPlanck’s constant. Equally troubling, though not as widely noted, is that we believe the universe isfinite, so if we could turn the universe into a vast computing device, or even under the assumptionthat it is such a device, there would only be a finite amount of space to write out the decimalplaces of any measurement.I claim that physics presents us with an epistemological problem with our conception of thefoundations of mathematics. Suppose I challenge you and say that despite everything you know,there is actually a largest natural number (in other words, a number of the form 0 , , ... ), though Idon’t quite have the facilities to show it to you. How do you prove to me that I am wrong? You Epistemological problems in philosophy ask about the nature of our “knowing” and how we know what we know.See, for example, the discussion in [17]. N , form N + 1, and it will be bigger.” However, in return I tell youthat to deliver N to you, it would take up all the storage in the universe, and, moreover, we wouldknow that I am representing N in the most compact format possible (e.g., perhaps I am usingiterated exponentiation, or even that I am using some of my space to define new symbols that areshorthand for some amount of repeated exponentiation and other things, and using all this to define N ). You have no ability to even talk to me about N + 1. Does this number “exist” or not? Youpush me further and say that you are talking about the number N + 1 in a “conceptual” way; afterall – every number has a successor – it is one of the postulates of Peano Arithmetic! But what doyou mean by such a conceptual way? For this N there is no mathematical way whatsoever to talkabout N + 1 (or any number larger than N for that matter). So the only way to talk about N + 1 isinformally – through our informal language. But even allowing informal language, given our finitevocabularies and space limitations there are only finitely many natural numbers we can describe inthis way, and again a maximum such number. While we might be able to get away with a phraseof the form “the largest natural number expressible using formal mathematics plus one” allowingthe phrase “the largest number we are able to express (informally) plus one” would be obviousrubbish. Thus, perhaps we must have a gentlemen’s agreement never to discuss anything beyondthis largest informally expressible number. Whether such a resolution is satisfactory (especially togentlewomen) I leave for the reader to decide.Let us now, however, circle back to formal mathematics. After all, an expression of the formlim n ! ∞ f ( n ), for some function f , is a formal mathematical expression, not something informal! Youmay say that as we take larger and larger n we are talking about going from n to n + 1 in a general sense, in the sense of n being any natural number, not in a particular natural number, let alonethe case of our special number N . For example, you say, I really don’t need to talk about N + 1in this particular instance since in all arguments about limits I am using some form of implicit orexplicit induction and the n in these arguments does not refer to specific values of n . For example,in the explicit induction proving that K X i =1 i = K ( K + 1)2 , (1)we establish the result for a generic n and then prove that it holds also for the generic n + 1. Thus,first stopping to observe that (1) holds for K = 0, we then assume it holds for K = n , in otherwords, n X i =1 i = n ( n + 1)2 . Then for K = n + 1 we establish n +1 X i =1 i = n X i =1 i + ( n + 1)= n ( n + 1)2 + 2( n + 1)2= ( n + 2)( n + 1)2 . n + 1 (and n + 2) in this “generic” sense. To this argument I say yes, whatyou have shown is indeed that the familiar expression (1) holds, at least for all practical values of K , and my intention is not to challenge all of existing finitary mathematics. However, let us nowreturn to the expression lim n ! ∞ f ( n ) and put L = lim n ! ∞ f ( n ). To a mathematician, such an expressionis equivalent to saying that for any arbitrarily small epsilon there is a point beyond which f ( n )is within epsilon of the limiting value L . In formal mathematics this notion is expressed thusly:( ∀ ǫ > ∃ m ( ∀ n > m ) | L − f ( n ) | < ǫ . Hence, the mathematician argues, for any given k one maythrow away the first k values and the limiting value, L , must be precisely the same. But what ifwe throw away everything up until our problematic N ? We end up with an induction that cannotget started! It is as if we have (1) but cannot even check the case K = 0.Lest you think this is a problem just of the arbitrarily large, note too that we have exactly thesame problem when talking about the arbitrarily small, for example, when taking any derivative, ortrying to evaluate any other expression of the form lim ǫ ! f ( ǫ ). The only difference in the case of thearbitrarily small is that we have to store, or otherwise represent, a potentially unlimited number ofdigits to the right of the decimal point. There is therefore a smallest definable ǫ > N .Physics therefore imposes a limit to the meaningfulness of mathematical assertions and weneed to question the sanctity of the edifice upon which much of classical mathematics is built.Despite the amazing structures that have been built atop it – constructs like Riemannian and semi-Riemannian manifolds, the backbone of our understanding of general relativity and gravity, HilbertSpaces of operators, the framework of our understanding of quantum mechanics, and many others,and all the wonderful theorems and further constructs that have been built atop this firmament– the foundations are in peril. Many a mathematician will bristle at this conclusion, since theaforementioned constructs speak to them like a great painting does to an art connoisseur, egginghim or her on with the beauty of the subject, as if to say that something so beautiful cannot possibly be built on an illusory edifice.If we leave our culturally acquired framework for understanding infinity and arbitrarily largenumbers at the doorstep, can we define these notions from more primitive ones in a way that willsatisfy the extreme finitist – someone who denies the ability to imagine arbitrarily large space orarbitrarily long time periods (and for whom an attempt to establish the former notions by assumingthese latter two seems circular)? The dilemma we now face is not dissimilar to that mathematicians had to confront at the turnof the 20th century, when Bertrand Russell unveiled his famous paradox that challenged the thenprevalent informal use of set theory. Prior to Russell’s discovery, mathematicians used the notion ofa set completely informally, and one could form collections of anything that mathematicians couldthink of and call the result a “set.” Russell asked us to consider the set N of all sets that do notcontain themselves (the set of all so-called “normal” sets). He then further asked whether the set N contained itself or not. If N ∈ N then N contains a set that contains itself, which is contraryto its definition. On the other hand if N / ∈ N then N does not contain all normal sets since N is one of them. One is thus led to a contradiction either way. There must, therefore, have been3omething wrong with the definition of N in the first place – and hence to our informal notion ofwhat it means to be a “set.”To get a sense of just how serious and unsettling Russell’s paradox was for mathematiciansof the day, note that its revelation came in 1901, just a year after Hilbert’s famous address atthe International Congress of Mathematicians where he unveiled his celebrated 23 problems, andproclaimed:“Take any definite unsolved problem, such as the question as to the irrationality of theEuler-Mascheroni constant, or the existence of an infinite number of prime numbers ofthe form 2 n + 1. However unapproachable these problems may seem to us, and howeverhelpless we stand before them, we have, nevertheless, the firm conviction that theirsolution must follow by a finite number of purely logical processes. ...This convictionof the solvability of every mathematical problem is a powerful incentive to the worker.We hear within us the perpetual call: There is the problem. Seek its solution. You canfind it by pure reason, for in mathematics there is no ignorabimus ” [5].In response to the Russell paradox, mathematicians created various axiomatizations of SetTheory - rules by which one could form new sets out of existing ones, beginning with a first set, theset with no elements, called the empty set. These axioms did not provide a means for creating aset of the type N encountered in Russell’s paradox, hence skirting that problem. The most popularaxiom system for Set Theory is today known as Zermelo-Fraenkel set theory with the Axiom ofChoice, a.k.a, ZFC [11]. But there is nothing completely “obvious” about the ZFC axioms, andthey have been hotly debated over time. Questions about the Axiom of Choice are legion, but thereis also the Axiom of Infinity that in an indirect fashion asserts the existence of infinite sets, andthe so-called Axiom Schema of Replacement – an infinite collection of axioms saying that given anyset X , and a function f defined on that set, then the image f ( X ) is also a set. Interestingly, theAxiom Schema of Replacement was not accepted by ZFC co-creator Ernst Zermelo [22] because itgives rise to the Downward L¨owenheimSkolem theorem of Model Theory, and the paradoxical resultthat there are countable (set) models of Set Theory. Of further note, Zermelo’s contributions toZFC were his attempt to reduce the so-called Well Ordering Principle, first conceived of by Cantoras a way to prove that any two sets have comparable cardinalities, to simpler principles – and, inparticular, to the customary statement of the Axiom of Choice – an axiom that Zermelo claimedto be obvious [3, p. 46].With the question of what it might mean to be a set in the back of our minds, let us circleback and ask a fundamental question: what is a real number? We have a sense of what a naturalnumber is. Say we are considering the number 3. We think of “3” as an abstraction for variouscollections of objects that can be put in one-to-one correspondence with one another and have thegiven number of distinguishable elements. But a real number – is it an abstraction for any thing orjust shorthand for some sort of approximate measurement that is getting more and more precise(or perhaps not) as we go, or an expansion of a well defined constant like π that, again, is gettingmore and more precise as we go? Is there any reality in the real number itself or is it just a way By the word ignorabimus , Hilbert was making implicit reference to the Latin phrase ignoramus et ignorabimus ,meaning “we do not know and will not know,” referring to the fact that scientific knowledge is inherently limited –an idea popularized by 19th century German physiologist Emil du Bois-Reymond. That given any collection of non-empty sets, one can form a new set by picking exactly one element from each ofthe sets.
4f summarizing a measurement or approximation activity ? Does mathematics in this sense blendirrevocably into physics?Mathematicians have almost uniformly dodged these questions, believing in the objective realityof mathematical concepts, like the different types of numbers and even the collection of all suchnumbers. Furthermore, in an effort to reduce everything in mathematics to one common foundation,following a precedent set by John von Neumann [20], the natural numbers and then the rationalsand the reals are typically created from sets. The number 0 is represented by the empty set ∅ = {} .Then one proceeds to construct 1 = { } = {∅} , { , } = {∅ , {∅}} , and so on, ultimately creatingthe natural numbers, N , the rationals Q , the reals R , the associated arithmetical operations, andbeyond – all simply by employing the axioms of ZFC. Whether this reduction of mathematics toSet Theory is natural or not is an open philosophical question. We have some intuition that variouscollections of three objects should be considered bona fide sets – but the number 3 itself? Are wenot reducing a concept that we have ironclad clarity about (i.e., the number 3) with something (aset) that we have just a very loose intuitive notion of? Ironically, the more intuitive notion most ofus have about the number ‘3’ being a conceptual shorthand for the collection of all sets with threeelements in them – a notion that can be made non-circular by depicting a single set with threeelements and then declaring ‘3’ to be the equivalence class of all sets equinumerous with the givenset – would lead us into trouble, at least in ZFC. If there were a set of all three element sets wewould easily be able to create a Russell-type paradox based on it.Before delving further into this subject, it is worth noting that even the most fundamentalbuilding block of our contemporary theoretical model of computation makes use of an infinitestructure – the Turing Machine with its infinite tape. The famous P=NP question is a questionabout limits and does not make sense without our ability to consider arbitrarily large integers. Wecannot distinguish P, the class of problems solvable in polynomial time, from EXP, the class ofproblems solvable in exponential time, in just a finite amount of space. Similarly for the HaltingProblem – all but the most trivial diagonalization arguments use at least countably infinite sets,as does finite model theory with its use of the Compactness Theorem of first order logic, i.e., toshow that there is no first order reachability predicate in graph theory . There is no question thatsome of these results can be salvaged. For example, we can ask whether it is possible to write aprogram in n bits of space that can tell whether an arbitrary program, also written using n bits ofspace, will halt. For n of “modest” size the answer is no and the argument is roughly analogous tothe classical relativized diagonalization argument used to prove the Space Hierarchy Theorem [16].However, when n gets close to the largest expressible natural number, the argument cannot becarried out so it is unclear whether we should accept the argument for “generic” n or not. Recently, the Swiss quantum physicist Nicolas Gisin, most well known for his experimental val-idation of Bell’s Inequality and quantum teleportation, has expressed concerns, much as I havearticulated them, in a series of papers [6–8] connecting problems with the foundations of mathe-matics with our conception of time and a constructive mathematical philosophy known as Intu-itionism [10]. A popular account of Gisin’s work in this area can be found in [27]. Intuitionismis a philosophy of mathematics that is in striking contrast to the Platonic Realism view adopted A result that can be extended to finite graph theory using Ehrenfeuch-Fra¨ıss´e games of arbitrary large sizes [13]– in this case at least not requiring any sort of completed infinite set.
5y most mathematicians [18, 19]. In the classical mathematical view, also known as mathematicalformalism, mathematical facts, or theorems, exist and just await our discovery. In the Intuitionisticview, on the other hand, mathematics is entirely a construct of the human mind. The Dutch math-ematician L.E.J. Brouwer, the founder of intuitionism and also a great classical mathematician,initially developed the subject from 1905-1910, thus predating all of contemporary computer sci-ence. Surely, however, there was nothing special in Brouwer’s thinking about the thought processof homo sapiens, so the thinking of other beings, or the in-memory constructs of computationalmachines, would also have been allowed had he done his work in the modern era. However, thedistinction between mathematics as facts to be discovered, and facts that can’t possibly exist untilthey are discovered by a mind, human or otherwise, is a very important one. On the one hand,this notion puts great emphasis on time. At the turn of the 20th century there was a great debatebetween Hilbert and Brouwer about the nature of the real numbers [8]. Hilbert promoted thenotion that every real number, with its infinite sequence of digits, was a completed object, whileBrouwer argued that real numbers just represented a never-ending process that develops over time in what he referred to as a “choice sequence.” Although Brouwer was supported in his view by thefamous mathematician, physicist and philosopher Hermann Weyl [21], and the great logician KurtG¨odel [9], Hilbert clearly won this debate and trained mathematicians of today are handed downa Platonic Realist view of real numbers.Gisin argues, as I have, that because a finite volume of space can only contain a finite amountof information, a real number must also only contain a finite amount of information. He suggeststhat the notion of a real number be replaced by what he calls a Finite Information Quantity [7].Suppose we are expanding a number x , between 0 and 1, and that we would ordinarily have a base2 expansion of the form x = b b ..., where each b i is a binary digit – a 0 or a 1. Like a Brouwer choice sequence, these values represent aprocess that develops over time. However, the numbers b i , for Gisin, are not bits in the conventionalsense, but each is a rational number in the closed interval [0 ,
1] giving the probability that theunderlying value is a 1. An underlying probability of 0 coincides with b i being definitively 0. Whenthe values are completely random, so that b i = 0 .
5, they contain no information. Once thesepseudo-bits are pinned down to a 0 or a 1 they contain the most information. The informationcontained in an entire Finite Information Quantity of this sort is given by I ( x ) = ∞ X i =1 I ( b i )= ∞ X i =1 (1 − h ( b i ))= ∞ X i =1 (1 + b i lg b i + (1 − b i ) lg(1 − b i )) . (2)At any point in time the b i are only ever specified up to some finite point, after which the b i areassumed to be completely indeterminate, with b i = 0 .
5, and hence the information content of allthese unspecified digits is nil. The sum (2) is therefore always finite. Where h (0) = h (1) = 0 by taking limits, in other words lim x ! x lg x = 0. | b i − . | ! i + 1)st digit would seem to be conditionally dependent on our certainty aboutthe i th digit, for all i . Moreover, the definition seems to leave unanswered the question of how, andto what accuracy, one is to specify the rational numbers that make up each b i . Finally, while thedefinition gives an appealing depiction of a limitation on the precision with which a a number canbe specified to the right of the decimal point, can we do something similar for very large numbers,in other words regarding the specification of digits to the left of the decimal point? Treating digitsto the left of the decimal point in the same way in which we have treated digits to the right of thedecimal point does not seem to work, but it does seem like we can treat the imprecision of a verylarge integer N simply as the imprecision in ǫ where N = ǫ , with ǫ as specified in (2).Returning to Brouwer, and the classical Intuitionistic understanding of mathematics, the notionof time is not just an aspect of the unfolding of digits of real numbers – the notion extends to allaspects of mathematics. A theorem starts out with premises and arrives at a conclusion – it isthe summary of a mathematical activity . It brings two not obviously equal things together, thecollection of mathematical objects satisfying the premises and those satisfying the conclusion, andsays they are the same. This activity can only be carried out as long as we have a notion of time– hence the view of mathematics as activity. Furthermore, from a modern perspective, a theoremis in part information. Information does not exist in the abstract. It must be materialized as bitssomewhere – in the head of a mathematician, on a written page, or stored on a computer, and thatmaterialization as bits requires time and space.Intuitionism is actually a deep and reasonably well studied subject. There is a conceptionof natural numbers and, as we have noted, of real numbers, though the conception of each isconsiderably different from the conventional ones. All infinite collections are considered “potentiallyinfinite” and special care has to be taken when reasoning about all natural numbers or all realnumbers. On the other hand, one can reason about any natural number or any real number. Thereis a principle of induction for the natural numbers, but a statement about the reals of the form“every bounded subset of reals has a least upper bound (in R )” runs into a problem.Although a further exploration of Intuitionism is beyond the scope of this essay, many questionsseem worth delving into: if we rescrutinize the foundations of mathematics and computer scienceunder the assumption that arbitrarily small and arbitrarily large quantities cannot exist, whathappens? What parts of the edifice fall apart and what parts can be salvaged? Does Intuitionismform a satisfactory foundation for a least practical computational reasoning? What parts of classicalcomplexity theory does Intuitionism not support (if any)? Can we develop complexity theory usingGisin’s notion of a Finite Information Quantity, though flipped, as we have indicated, so the notioncan apply to large integers? Finally, are these questions philosophical ones that will never impactthe practice of science (and so, perhaps, best be speculated upon at a local pub), or are they, rather,deep scientific questions that could impact our understanding of the universe and even impact therole science has in informing the practice of applied science and engineering? I shall delve intothese matters with a bit more precision and the specification of three questions for further researchat the very end of this essay. 7 Ultrafinitism
The notion that there may be a largest natural number has been argued in other ways by variousrepresentatives of a school of mathematics known as Ultrafinitism [25]. Like Intuitionists, Ultrafini-tists do not accept the notion of a completed infinite set. Some Ultrafinitists do not accept naturalnumbers that cannot be expressed in a binary or other decimal expansion that would require morestorage space than the assumed storage space of the universe. Others take issue with certain explicitsymbolic expressions of natural numbers that they view as potentially uncomputable e.g., takingthe floor ( ⌊ x ⌋ ) of a potential upper bound for the first Skewes Number that is dependent on theRiemann Hypothesis: [24] e e e . Skewes numbers come up in number theory because they are the crossing points of the primecounting number π ( x ), the function giving the number of primes less than a given number x ,and the logarithmic integral function, li( x ) = R x dt ln x , which, by the Prime Number Theorem,approximates the same number.Closer to the spirit of this essay is the attempt by Kornai [12] to characterize the set of num-bers expressible via the extraordinarily fast growing Ackermann functions in a theoretical universecapable of storing 2 bytes. 2 bytes ≈ bits, is somewhat more than the computationalcapacity of the universe, estimated at 10 bits in [14].A compelling view of Ultrafinitism has been expressed by the Rutgers mathematician DoronZeilberger [28], although he has not written a great deal on the subject. Of note is Zeilberger’sobservation that integral calculus can be developed perfectly well without limits, and in fact wasdeveloped in a much more informal way by its creators Newton and Leibniz in the late 1600s. Theepsilon-delta formulation that mathematicians of today are familiar with was not developed untilthe early 1800s by Balzono and Cauchy [4]. In Zeilberger’s view a differential equation is just themeaningless “degenerate case” of a finite difference equation.An important contribution to Ultrafinitism is the work on so-called Predicative Arithmetic dueto Princeton mathematician Ed Nelson [15]. Nelson found a certain incompatibility in the Peanoaxioms, with the unconstrained axiom schema of induction and the essentially constructivist axiomsdefining the properties of the successor function. In his Predicative Arithmetic inductive argumentscannot be “predicated” upon numbers that have not yet been shown to exist. Thus the notion ofa natural number being even can be defined perfectly well as a natural number n for which thereis a previously constructed natural number m and n = 2 m . However, claiming, say, that there isa number divisible by every prime between 1 and n , at stage n , is not valid [26]. It is conceivablethat the notion of predicativty has some connection to the P=NP problem (e.g., in distinguishingthe languages in P from those in NP). As I end this essay I think it is appropriate to offer two quotes about time from Brouwer and Ein-stein, although the quote by the former will sound dated. I will then wrap up with three of the mostpressing questions for future research that arise from our discussion. As I’ve noted, time is centralto Brouwer’s notion of Intuitionism – a theorem does not make sense outside of time. Brouwer’sIntuitionism broke sharply with Hilbert’s formalistic school of mathematics [23] (in fact the two8en never got along, and had a much publicized falling out in the 1920s over editorial policies atthe emminent journal,
Mathematische Annalen ). Hilbert considered mathematics and logic to bestatements about the consequences of the manipulation of sequences of symbols using establishedmanipulation rules. His view was that the symbols could be stripped of any “interpretation” andbe dealt with purely syntactically. His aim was to develop a complete and provably consistentaxiomatization of all of mathematics. He was the first to articulate how to attempt to do so usinga formal language. According to Hilbert, the language must include five components [23]: • It must include variables such as x, which can stand for some number. • It must have quantifiers such as the symbol for the existence of an object. • It must include equality. • It must include connectives such as ↔ for “if and only if.” • It must include certain undefined terms called parameters. For geometry, these undefinedterms might be something like a point or a line, which we still choose symbols for.Of course G¨odel’s Incompleteness Theorem(s) dealt a death blow to Hilbert’s Program. Althoughthe original goal of establishing completeness and consistency has been abandoned, the vast majorityof contemporary mathematics still follows Hilbert’s paradigm.Brouwer, looked at mathematics very differently. He did not view axiomatization or even thelanguage of mathematics as paramount. To Brouwer, axiomatization and language, as Hilbertconceived them, were attempts to introduce rigor into an inherently intuitive process. Brouwerdeclared two founding conceptual “Acts of Intuitionism” that gave rise to his various theories. Thefirst of these acts was the following:“
The First Act of Intuitionism.
Completely separating mathematics from mathematicallanguage and hence from the phenomena of language described by theoretical logic,recognizing that intuitionistic mathematics is an essentially languageless activity of themind having its origin in the perception of a move of time. This perception of a moveof time may be described as the falling apart of a life moment into two distinct things,one of which gives way to the other, but is retained by memory. If the twoity thus bornis divested of all quality, it passes into the empty form of the common substratum ofall twoities. And it is this common substratum, this empty form, which is the basicintuition of mathematics.” – L.E.J. Brouwer [1, pp. 4–5]Time also plays a central role in the special and general theories of relativity on account of theLorentzian space-time continuum and the warping of space-time that is the manifestation of thegravitational force. The following quote is from a letter Einstein wrote shortly before his death in1955, upon the passing of his life-long friend and scientific sounding board, Swiss/Italian engineer,Michele Besso. The letter was to Besso’s family.“People like us, who believe in physics, know that the distinction between past, presentand future is only a stubbornly persistent illusion.” – Albert Einstein [2]Although, if we take the Intuitionists’ word for it and agree that mathematics is inescapablya mental activity intertwined with time, we are brought back, as Einstein was, to the question of9what is time?’ In light of the relativity of simultaneity there is no well defined present moment,and to someone traveling at the speed of light relative to us, because of time dilation, our lifetimesgo by, to them, in the blink of an eye.Let me now conclude with the three promised questions for future research:(1) If we make the bit and the symbols and the foundational construct of all of mathemat-ics, replacing the set and the associated symnbols { and } , how should development of thefoundations proceed? Should the process be axiomatic or do we take the extreme stance ofBrouwer and dispense with an axiomatic approach? Whether or not we side with Brouweron axiomatics, how does one recover the notion of a set from that of a bit? Should the notionof a set instead just be part of the logical “plumbing?” If we try to take Gisin’s approachand think of real numbers as finite information quantities it seems that we need a notion ofprobability before we have a notion of real or other numbers. How can such an idea be madeto work ‘ground-up?’ If we consider making the qubit the foundational concept rather thanthe bit we run into the same problem.(2) Are Cantor’s diagonalization argument, and complexity questions such as P=NP, really mean-ingful? Note that our ability to separate complexity classes almost completely relies on diag-onalization arguments. There is obviously no question that the classical asymptotic analysisof our most common algorithms, say for sorting or finding shortest paths, is meaningful andimmensely practical. If we are to assert that P=NP is not meaningful, we need to do so veryprecisely and in such a way as to not throw out the baby with the bath water.(3) In our discussion of inexpressivity, can we be completely precise about the smallest numberinexpressible using n bits? After all, there are theorems about such inexpressivity in finite(as well as infinitary) model theory. For example, reachability in undirected graphs is knownto be inexpressible in the 1st order logic of graphs but expressible in the monadic 2nd orderlogic of graphs [13]. If we specify a language L , say the first or second order language ofPeano Arithmetic, then the smallest number inexpressible in n bits in L will be expressiblein n bits in a more expressible language L – a language that can express expressibility in L (though not its own internal expressibility/inexpressibility). We thus have L ⊃ L . Wecould seemingly continue in this fashion, obtaining languages · · · ⊃ L ⊃ L ⊃ L . The factremains, however, that there is only finitely much we can express with n bits and so at somepoint we cannot continue the process of taking language extensions. At what point is therea breakdown and how do we state this conundrum formally?A natural argument to try to make an end run around these inexpressivity limitations is toincorporate time in addition to space. In other words, at time t = 0 we define a largest (orextremely large) natural number N using most, or all, of our storage. Then at time t = 1we give a name to N , and reuse our storage to define a much larger number N , and soon, obtaining ever larger numbers in time. Since we are assuming finite space, there is thenthe obvious objection about finite time. However, there is a more serious objection as well.Suppose we have created such an N k for k ≥
1. Since we have destroyed all record of howthe N , ..., N k − were created our N k is in fact meaningless. Also known as s - t connectivity, in other words that there is a path between two named vertices. eferences [1] L. E. J. Brouwer. Brouwers Cambridge lectures on Intuitionism . Cambridge University Press,Cambridge, UK, 1981.[2] A. Einstein. Times arrow: Albert Einsteins letters to Michele Besso. ,2020. [Online; accessed 5-May-2020].[3] A. B. Feferman and S. Feferman.
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Kurt G¨odel Collected Works Volume IV Correspondence A-G . “Choice sequencesare something concretely evident and therefore are finitary in Hilberts sense, even if Hilberthimself was perhaps of another opinion.” p.269, Oxford University Press, Oxford, UK, 2003.[10] R. Iemhoff. Intuitionism in the philosophy of mathematics. In E. N. Zalta, editor,
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