Reality

Infinity: What Is It (And Where To Find One Of Your Own)

The Imagination

Infinity is big. In fact, it is very, very big. It is even bigger than the incalculable Graham’s number. Now Graham’s number is big. I mean it is really, really big. It even makes Douglas Adam’s view of the universe look small!

Graham’s number is unimaginably larger than other well-known large numbers such as a googol, googolplex, and even larger than Skewes’ number and Moser’s number. Indeed, the observable universe is far too small to contain an ordinary digital representation of Graham’s number, assuming that each digit occupies at least one Planck volume. Even power towers of the form  are useless for this purpose, although it can be easily described by recursive formulas using Knuth’s up-arrow notation or the equivalent, as was done by Graham. The last ten digits of Graham’s number are … 2464195387″ (1)

Let’s put the brakes on for a moment before we start tripping out. There are a few things about numbers that we need to get straight before we continue into the imaginary realms of infinity.

A question for you: what is the biggest number you can imagine? Not in terms of number, but in terms of imagined things. This is an important reminder because the real thinking powers of the average Western educated European is actually very poor. That is, after all, why I write all this nonsense.

This is infinite to many imaginations.

So try imagining seven hats, and eight – nine, ten and twelve. Twenty five? Fifty?

… A hundred?

I can get that far, and I am well beyond the imaginative powers of most people who will ever read this. The usual number is seven or eight (2).

That is tiny!

Yes, it is tiny. That is how little we can really conceive of infinities. It also shows that our ability to imagine the nature of infinities is equally weak. Imaginary notions like parallel universes, and infinite numbers of monkeys typing on infinite numbers of typewriters are but two (3).

Straight Talk About Infinities

This isn’t just silly talk, this is straight up. No nonsense. If you want to start dealing with infinities, you really have to get your imagination into gear. Then you have to start thinking. Not just wishy-washy logic, where push equals shove. We are dealing with realms that are really powerful. The real concept of infinity is so powerful it really would blow your mind.

Infinities are actually all around us. Remember that an effective infinity is beyond your power to imagine it in “real” terms – that is to say more than 12 hats. Sure you can count beyond that, but that is just repetition, computer like rote counting. That is the sort of thing you leave to a machine. It isn’t called number crunching for nothing! Infinities are really about the imagination. Because we often forget the infinitessimal infinities which are equally untouchable by most imaginations. Everyone thinks of atoms, think they exist. But do they, really? Really, really? So they do, right? (4) Show me one and I will show you an infinitessimal infinity. Of course, there is the infinitessimal of the infinitessimal. Again, I digress.

I want to cut to the bone. Numbers are not all they seem. Benford’s law should have shown you that! Numbers and digits possess powers that bend the mind, nevertheless, they exist. In part this is because numbers are actually a human construct. They are a way for humans to come to terms with the world around us. I am working on a piece about how the ancient Egyptians dealt with this because they didn’t use numbers as such. Not at least when it came to things like the pyramids. They used imaginative faculties that elude even me, and when it comes to imagining things, you would have a hard time out pacing me. Mind you, we could have some real fun trying.

Catching An Infinity Of Your Very Own

A piece of European A4 paper,
its length is the diagonal of
a square the size of its shorter side.

I want to bring infinity back home where it belongs. In fact you will have a few of them on your desk at work, and possibly next to your computer. Certainly in your filing cabinet, or wherever you keep your documents. I am speaking of pieces of paper. The humble A4 sheet of paper is stuffed full of infinities of every size and dimension. Putting aside the number of atoms it doesn’t comprise of (5) – which is a fixed number anyway – it is the ratios inherent in the relationship between the two sides. The square root of two, which is the relationship concerned is a so-called irrational. That means it cannot be calculated using conventional decimal notation. Because this is your own pet infinity, and you have a pack of them and they cost you three euros! Not bad, eh?

You can have a look at the square root of two to ten thousand decimal places and here it is! (Or if you are a sensible person, you’ll just skip the meaningless numbers).

1. 4142135623 7309504880 1688724209 6980785696 7187537694 8073176679 7379907324 7846210703 8850387534 3276415727 3501384623 0912297024 9248360558 5073721264 4121497099 9358314132 2266592750 5592755799 9505011527 8206057147 0109559971 6059702745 3459686201 4728517418 6408891986 0955232923 0484308714 3214508397 6260362799 5251407989 6872533965 4633180882 9640620615 2583523950 5474575028 7759961729 8355752203 3753185701 1354374603 4084988471 6038689997 0699004815 0305440277 9031645424 7823068492 9369186215 8057846311 1596668713 0130156185 6898723723 5288509264 8612494977 1542183342 0428568606 0146824720 7714358548 7415565706 9677653720 2264854470 1585880162 0758474922 6572260020 8558446652 1458398893 9443709265 9180031138 8246468157 0826301005 9485870400 3186480342 1948972782 9064104507 2636881313 7398552561 1732204024 5091227700 2269411275 7362728049 5738108967 5040183698 6836845072 5799364729 0607629969 4138047565 4823728997 1803268024 7442062926 9124859052 1810044598 4215059112 0249441341 7285314781 0580360337 1077309182 8693147101 7111168391 6581726889 4197587165 8215212822 9518488472 0896946338 6289156288 2765952635 1405422676 5323969461 7511291602 4087155101 3515045538 1287560052 6314680171 2740265396 9470240300 5174953188 6292563138 5188163478 0015693691 7688185237 8684052287 8376293892 1430065586 9568685964 5951555016 4472450983 6896036887 3231143894 1557665104 0883914292 3381132060 5243362948 5317049915 7717562285 4974143899 9188021762 4309652065 6421182731 6726257539 5947172559 3463723863 2261482742 6222086711 5583959992 6521176252 6989175409 8815934864 0083457085 1814722318 1420407042 6509056532 3333984364 5786579679 6519267292 3998753666 1721598257 8860263363 6178274959 9421940377 7753681426 2177387991 9455139723 1274066898 3299898953 8672882285 6378697749 6625199665 8352577619 8939322845 3447356947 9496295216 8891485492 5389047558 2883452609 6524096542 8893945386 4662574492 7556381964 4103169798 3306185201 9379384940 0571563337 2054806854 0575867999 6701213722 3947582142 6306585132 2174088323 8294728761 7393647467 8374319600 0159218880 7347857617 2522118674 9042497736 6929207311 0963697216 0893370866 1156734585 3348332952 5467585164 4710757848 6024636008 3444911481 8587655554 2864551233 1421992631 1332517970 6084365597 0435285641 0087918500 7603610091 5946567067 6883605571 7400767569 0509613671 9401324935 6052401859 9910506210 8163597726 4313806054 6701029356 9971042425 1057817495 3105725593 4984451126 9227803449 1350663756 8747760283 1628296055 3242242695 7534529028 8387684464 2917328277 0888318087 0253398523 3812274999 0812371892 5407264753 6785030482 1591801886 1671089728 6922920119 7599880703 8185433325 3646021108 2299279293 0728717807 9988809917 6741774108 9830608003 2631181642 7988231171 5436386966 1702999934 1616148786 8601804550 5553986913 1151860103 8637532500 4558186044 8040750241 1951843056 7453368361 3674597374 4239885532 8517930896 0373898915 1731958741 3442881784 2125021916 9518755934 4438739618 9314549999 9061075870 4909026088 3517636224 7497578588 5836803745 7931157339 8020999866 2218694992 2595913276 4236194105 9210032802 6149874566 5996888740 6795616739 1859572888 6424734635 8588686449 6822386006 9833526427 9905628316 5613913942 5576490620 6518602164 7263033362 9750756978 7060660685 6498160092 7187092921 5313236828 1356988937 0974165044 7459096053 7472796524 4770940992 4123871061 4470543986 7436473384 7745481910 0872886222 1495895295 9118789214 9179833981 0837882781 5306556231 5810360648 6758730360 1450227320 8829351341 3872276841 7667843690 5294286984 9083845574 4579409598 6260742499 5491680285 3077398938 2960362133 5398753205 0919989360 7513906444 4957684569 9347127636 4507163279 1547015977 3354863893 9423257277 5400382602 7478567417 2580951416 3071595978 4981800944 3560379390 9855901682 7215403458 1581521004 9366629534 4882710729 2396602321 6382382666 1262683050 2572781169 4510353793 7156882336 5932297823 1929860646 7978986409 2085609558 1426143636 3100461559 4332550474 4939759339 9912541953 2300932175 3044765339 6470662761 1661753518 7546462096 7634558738 6164880198 8484974792 6404506544 4896910040 7942118169 2579685756 3784881498 9864168549 9491635761 4484047021 0339892153 4237703723 3353115645 9443897036 5316672194 9049351882 9058063074 0134686264 1672470110 6534634939 1640714628 5567980177 9338144240 4526913706 6609777638 7848662380 0339232437 0474115331 8725319060 1916599645 5381157888 4138084332 3210533767 4618121780 1429609283 2411362752 5408873729 0512940733 9479433061 9439569367 0207942951 5878228349 3219316664 1113015495 9469837897 7674344435 3933770995 7134988407 8908508158 9236607008 8658105470 9497904657 2298888089 2461282816 0131337010 2908029099 9745647849 5815456146 4871551639 0502419857 9061310934 5878330620 0262207372 4716766854 5549990499 4085710809 9257599288 9323661543 8271955005 7816251330 3815314657 7907926868 5008069844 2847915242 4275441026 8057563215 6532206188 5751225113 0639370253 6292716196 8251259192 0252160587 0118959673 2244239267 4237344907 6464672737 5347964598 8191498079 3171800242 3855453886 0383683108 0077918246 6462754117 4442500187 2777951816 4383451463 4612990207 6334301796 8554385631 6677235183 8933666704 2222110939 1449302879 6381283988 9311731308 4300421255 5018549850 6529455637 7660314612 5590910461 1384768282 3595924772 2862904264 2736163264 5854433928 7726386034 3149804896 3973633297 5488592568 1149296836 1267258985 7383321643 6663487023 4773026101 0613050729 8611534129 9488087744 7311122954 2652751653 6659117301 4236062652 5869077198 2170370981 0464436047 7226739282 9874152593 0695620638 4710827408 2184906737 2330587430 2970924289 9481739244 0786937528 4401044399 0485208788 5191419354 1512900681 7351703069 3869705900 4742515765 5248078447 3621441050 1620084544 4122255956 2029847259 4035280190 6798068098 3003964539 8568593045 8625260637 7974535599 2774729906 4888745451 2424960763 7801086390 0191058092 8747647207 5110923860 5950195432 2816020887 9621516233 8521612875 2285180252 9287618325 7037172857 4067639449 0982546442 2184654308 8066105802 0158472840 6712630254 5937989065 0816857137 1656685941 3005331970 3659640337 6674146104 9563765103 0836613489 3109478026 8129355733 1890551970 5201845150 3996909866 3152512411 6111925940 5528085649 8931958983 4562331983 6834948808 0617156243 9112866312 7978483719 7895336901 5277600549 8055166350 1978555711 0140555297 6338412750 4468604647 6631832661 1651820675 0120476699 1098721910 4447440326 8943641595 9427921994 4235537187 0429955924 0314091712 8481585438 6600538571 3583639816 3094524075 5700932516 8243441682 4083619792 7337282521 5462246961 5332170268 2995097908 9034594858 8783494396 1620435842 2497397187 1139589273 0509219705 4917176961 6004455808 9942787888 0369169432 8945951472 2672292612 4850696173 1638094108 2186004528 6102696547 5763043102 5602715231 3969482135 5198214097 1654909731 9992834925 6740974903 9229712634 8693414574 9331980417 1807611196 3902278664 0759224341 6776246623 6238913110 2703433045 7636814112 8321326308 5822394562 1959808661 2939996201 2341561763 1817431242 0089014983 8485604808 7986460839 3596492366 5142968125 7731432291 4568716827 6219961182 7826953157 4983802624 6517590541 0397618128 7604216386 1345022132 6272775661 2441133610 7751955577 4950865636 0673786650 6231856406 9912280187 5741785494 6612532759 9769796059 7760590756 4891066610 1583841720 2818530432 1190446577 5255427754 3798726054 8817361982 6758168628 3295260789 9322266836 0283851351 2281059318 5910286415 0815705631 9717315183 1362502435 9041463212 2392176633 9826893682 5315053005 9891547029 0953719326 6207341123 4947433678 8469020139 0497842852 1634144292 1458955828 7847669394 6464267812 2190497856 3635526336 8278051860 0986992489 3778600239 8769169807 6566219438 9854437080 5946433362 3338105874 5816235475 6001365924 3524265714 3083465545 7680023708 1467573252 5470255074 7637471635 0678515991 7369379325 1032682760 6286459146 1820472148 6370370771 9269268236 2333472037 9245964691 8105261391 5308628029 1440965482 5638730927 3042654466 2929045896 0637519187 1146934536 1973324789 5727070315 3093090192 1199199993 6157650035 0398405406 7425387927 5279227247 3356677060 7837911384 4889362613 6765706026 3600315132 9520953952 0285489738 4486256134 9244147086 0708660267 6349978793 4208758361 2194711699 4223848482 5959143045 2810706260 1508969135 3030177200 6271705440 2090669514 9152745977 1970594769 5474095210 2878725578 5688002219 3717743558 1107939308 8338455864 8277291008 6295545661 4130672123 0848740227 1210586863 2338823741 3884428938 1554446471 0575565146 8435702946 6350628938 7356986868 8376480326 5195284146 5351739530 2736120137 4203009867 3983851432 1900436028 9826982935 2939941412 9230580384 5650227072 1681516194 1011449826 3013649008 7704839848 8386090653 3685990545 8389520318 5648041493 2721423908 6516499943 1659207965 9535694307 2311291162 9286797517 1566889054 3932203569 1293324570 2080671944 4049730494 3981408227 8296027994 2454108316 6675921424 8351827238 1720504103 9274288801 5562233807 9614751243 3514731021 2845459448 9944499600 0752437519 5701166834 1744749079 5882099517 8367680232 3651767497 2301487457 7427259947 6096219843 2714835298 6111902728 7358490521 7975908374 1974860267 0605374623 1530039375 2123678677 5284869219 5857137554 2696848278 3631786110 9933680143 9159059748 4285805451 6130230143 9790570161 0889862777 9610750673 3326760486 5492925139 9781390535 8822768937 3220494148 3940135560 3565604421 4017612060 5131806891 9899626061 8483185340 1836237821 7266375804 5524719626 6174925422 8528045714 4204857834 2113228008 5287042054 8899234127 8554812367 6153770710 4254469868 5219911228 3542663499 9712748366 0762462418 2073646661 7128394748 4732804744 3040334410 7200428727 1275670279 5675824292 6271945458 0530026664 8996507956 9778178621 9421720052 3716536946 7704195111 9127046248 3605113028 9046437751 1486948878 4961511884 1471910001 2558838366 6067720841 1235153558 8112677895 7155859041 2576261601 0675131535 8021242733 1871000635 8249545040 9957940725 4798900316 8265123731 1905566829 1519430537 0848930786 9197428290 4903860372 3116099283 4243171222 5099454715 0192866648 7871079519 9518005463 3883844315 4817246354 8024451803 0845273431 0006213710 3462573306 0012349737 4435581809 6567846464 1533905146 5691932456 2353140577 9193698988 4236471835 2537580525 7713311200 7971040683 1549266540 2026046806 8183914378 2721476906 3242469517 1286367384 4313983337 1176159418 6999346626 2345373452 3567940124 1680922911 6360956372 1674528391 7099091466 4850739205 1516056047 3787106154 7021699607 4656930979 4426121469 2561593425 6494019122 9895147325 4471518126 3258368897 2822628332 9524035970 0727863364 6045947071 2417472946 8775705958 1573499628 4809956783 9255474240 4489918870 7106967524 2507745201 2293608105 7414265323 4724064162 1410333533 4055110452 1261750359 0284037454 5918645047 2762434207 1770929793 5401021409 6464502836 8341804075 8608100140 7216192477 1798098596 8111540446 4437285689 5928683197 7797786934 6415984697 4513391774 1537904877 8808300220 5833504674 6555323028 5873258352 … (6)

It isn’t that interesting, is it? How many of you actually read all of those numbers? I certainly didn’t! Remember that by the time you get to the fourth decimal place you are already well beyond the realms of the sensible. Ten thousand places is for mathematicians and the autistic. Oh, and of course our pet computers. In decimal terms, the ongoing stream of numbers is unstoppable. That is truly infinite – and it’s sitting by your side as you work. And all these decimals mean absolutely nothing when brought into the light of day.

Wait! Take a peek at the photos above that I made earlier in real Blue Peter style. The ratio in the piece of paper is there. It just is. So what is going on here? How is it that something so simple can be so complex??? The fact is that it is we that have made it complex by trying to squeeze it into the intellectual realm. That is to say, modern mathematics that is dependent on decimal notation. I have already demonstrated in my piece about Benford’s law that digits are anything but ordinary (7). This exceptionally powerful tool, mathematics, is in fact part of the problem we face. It is trying to deal with fluid, dynamic things with ideas and notions that are anything but. Even if we started counting in base “root two” (not as impossible as you think, by the way for we have natural logarithms) we would encounter the inverse problem, namely that ordinary numbers would now be incommensurate! There are infinities we just can’t escape … !

Hang on just one moment!

Dealing With Real Infinities In Real Terms

How is it that a piece of paper can just exist, when it contains infinities? Well, it just does. Just because you impose some kind of abstract notation to it, does not mean that those infinities actually exist. That in our age of debt bombs and hyper inflation is a very real problem indeed. The academic approach to fixing the woes of the planet is simply to do more of the same, after all, there is no other way to think, is there? Well you are wrong. Sorry. There are other ways, and believe me they are as powerful as any infinity. That is because they can deal with them directly.

I want to cut to the bone of the issue, I didn’t quite manage it the last time: I have skirted around long enough, and to be honest I am sure that I will have already lost a few of you on the way as it is. An infinity is an abstract notion that is frankly absurd. It has little real use in our lives. The life of the ordinary western European is limited to around eight or ten, the amount you can really imagine unassisted. That is a very small number when compared to your weekly income! Let alone the price of the home you live in. Numbers are really Pandora’s butterflies let out of the box, and have wrought terrible damage. So what is the reality? The reality is the piece of A4 paper next to you. The relationship is there, it is real and tangible. You can trace the sides with your fingers right now. There is in fact nothing infinite about it at all. Those infinities are entirely notional, absurd and ridiculous. Bring some reality into your life, forget the square root of two and look not at the numbers, but at the beauty of the relationship instead. That is infinitely more powerful!

Footnotes:

 

(1) More about Graham’s number here. A googol is ten raised to the power of 100 – it is as abstract as it gets!! That is to say

10,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000

Yes, those two Stanford academics got it wrong! Google was named for the googol number, for reasons you will readily appreciate. They tried downloading the internet in 1992 – it was big enough then to crash Stanford University’s computers. They needed something else. And so it became Google instead of Googol. Mind you, doesn’t Google have a better feel to it?

(2) What is it about innumerate African goat herders
that allows them to notice things we can’t?

(3) Black and White: The Limitations Of Logic. There is a logical proposition that if you take an infinite number of monkeys and tie them to an infinite number of typewriters for an infinite number of years, you will eventually have the complete works of Shakespeare. Which of course is absurd. It is the illogical consequence of over-egged logical thinking. The logic has one flaw that sets off an explosion of consequential errors. First I would like to ask anyone daft enough to be reading this one question: how much time would it take to read through all those pages to find the correct typescript? How many times would you come across Joyce’s “Ulysses” with one spelling error? How many times would you have the complete works of Oscar wilde or snatches of Jane Austen mixed into the Bible? In amongst pages and pages of complete and utter nonsense.

(4), (5) Do atoms really exist??

(6) The square root of two to ten thousand decimal places

(7) A little about Benford’s Law

. And why it is important. Really, really important!

This is Benford's law. If law you can call such a thing. It is one of those laws that has no fixed physical basis.

Well Benford was an interesting chap. He noticed things that others missed. He wasn’t the first, and he won’t be the last. But he is the one who is remembered for this, and so it is called after him in the time appointed way. He noticed what a few others had also seen: that the logarithm tables at his local library showed more use at the front than at the back. Being that sort of a chap, this raised his interest. After all, if you notice these things, then you become inquisitive.

 

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