{"id":23185,"date":"2018-12-31T23:59:31","date_gmt":"2018-12-31T12:59:31","guid":{"rendered":"http:\/\/www.campaignmastery.com\/blog\/?p=23185"},"modified":"2018-12-31T19:11:58","modified_gmt":"2018-12-31T08:11:58","slug":"sizes-of-infinity","status":"publish","type":"post","link":"https:\/\/www.campaignmastery.com\/blog\/sizes-of-infinity\/","title":{"rendered":"Sizes Of Infinity"},"content":{"rendered":"

\nNew years are about new beginnings, about punctuating the year that was, to separate it (however artificially, however optimistically) from the year that is to come. This article is about new beginnings, and being aware of the options you have, but it might not seem so, at first. Bear with me….\n<\/p><\/blockquote>\n

One of the hardest concepts in mathematics for a lot of people to wrap their heads around is the maths of infinity. That’s because infinity is weird. No, I mean really<\/em> weird.<\/p>\n

Let’s start with an easy one:<\/p>\n

\"\"
\ni.e. any number multiplied by infinity gives infinity.<\/em><\/p>\n

Infinity is already infinitely large, it can’t get any bigger.<\/p>\n

But if you divide both sides of that equation by infinity, you get “any number equals one”, and that makes no sense whatsoever. The only alternative is to decide that infinity divided by infinity gives infinity!<\/strong><\/em><\/p>\n

How about this:<\/p>\n

\"\"<\/p>\n

i.e. any number added to infinity is still infinity.<\/em><\/p>\n

This happens because infinity is not just the biggest number that you can think of, it’s – literally – infinitely larger than that. It’s easy to show that logically, that also means that infinity divided by any finite number is still infinite.<\/p>\n

A little standard algebra turns the last equation into:<\/p>\n

\"\"<\/p>\n

i.e. it doesn’t matter how much you take away from infinity, there’s still an infinite amount left over.<\/em><\/p>\n

In fact, you can subtract any finite<\/em> number from infinity without making a dent. Actually, you can do so infinite times<\/em> and you’ll still<\/em> be left with infinity.<\/p>\n

A little thought turns that into:<\/p>\n

\"\"<\/p>\n

– not<\/em> zero as would be the case when subtracting any finite<\/em> number from itself (are you starting to get a glimpse of how weird a concept “infinity” actually is, yet?)<\/p>\n

There’s only one way to sum up that last equation: all infinities may be infinitely large but they are not<\/em> the same size. And with that, we’re through the looking-glass.<\/p>\n

An Empty Reflection?<\/h3>\n

Some might be tempted to say that nonsense results just mean that “infinity” doesn’t really exist! Certainly, physicists get really uncomfortable whenever infinity symbols start turning up in their calculations, because they have never been able to come to a consensus on the issue.<\/p>\n

My first year course in calculus at university had a fair amount of focus on “trends in dependent variables described as a function of an independent variable, as the value of that independent variable approached infinity”, something that I mentally summarized as the “ultimate trend”. This turns out to have all sorts of applications in the real worlds of physics, biology, and geology (amongst many others), as well as being of interest mathematically.<\/p>\n

That must mean that “the trend as x approaches infinity” must mean something real, and so infinity itself has to have some<\/em> sort of real meaning, even if it’s just an abstract ideal like “Absolute Zero”.<\/p>\n

Which means that infinity is real, and all those head-scratching calculations are real. “But surely you can’t prove the suggestion that infinities are different sizes?” comes the last gasp of sanity, fighting pluckily over in the corner. Well, let’s see.<\/p>\n

\"\"

plot to same scale of y=one half of 2 to the power of x, y=x, and y=the square root of x, respectively.<\/p><\/div>\n

Here are the graphs of three different mathematical functions. We all know how these work – you pick a number on the horizontal (x) axis, go straight up or down until we find the curve, then straight left or right to the vertical or (y) axis to read off the result. All three of these trend toward y=infinity as x approaches infinity.<\/p>\n

If you plot all three functions on the one graph, to the same scale, you soon find that three points are automatically defined: 1,1 is the only spot where all three cross one another, two of them also converge on 0,0, while the third crosses the y axis at 0,0.5. That gives two points for each of the curves, which in turn enables them to be scaled and positioned perfectly.<\/p>\n

In fact, the equivalence defines a boundary region – at any point in between an x of 0 and an x of 1, a real result can be obtained using the function. Outside of that range, it’s not so certain. The third curve simply doesn’t exist for values of x less than zero, and the first keeps getting closer and closer to a y of zero but never quite gets all the way in that region. But it’s the region to the right on the x axis of this defined point that is of most interest, because somewhere on that line is the “point” of infinity, at which point the function result is the trend as x approaches infinity.<\/p>\n

So, if I put all three graphs together like that, and put a mark on the x axis greater than one and say “this represents infinity”, you get this:<\/p>\n

\"\"

the same three functions on the one graph<\/p><\/div>\n

And, since we already know that the results of all three functions are infinity at the point where x=infinity, you can immediately see that all three give “infinities” of different sizes. And, in fact, you can confirm that by making another mark even further to the right, also labeling it infinity, and looking at the resulting<\/em> infinities. You could almost say that “infinity greater than infinity” is both true and false at the same time – but that’s not quite right; the problem is that you can’t glance at them and say one is larger than the other, you need something else to give them context. So it’s more accurate to suggest that, like Schrodinger’s Cat and other quantum states, the outcome is simply unresolved.<\/p>\n

You can even derive, from this set of functions and the infinities that result, all those calculations of infinity that I described earlier – proof that I’m not making this stuff up. Infinity is real,<\/em> and really strange.<\/em><\/p>\n

Infinite Combinations In RPGs<\/h3>\n

Most GMs and game designers never spend time thinking about infinity. And that’s a mistake, because the mathematics of infinities are vital<\/em> to running an RPG.<\/p>\n

Let’s say that you decide to run an RPG Campaign. You have an infinite array of possibilities in plot and character, and the first thing you have to do is cut that infinity down to size.<\/p>\n

You start by choosing a genre, or a game system, which defines a genre for you. That takes away the infinite number of options that are unique to all those other genres. So, what you’re left with is infinitely smaller than the infinity that you started with – but it’s still infinite. You can never<\/em> subtract a number from infinity and be left with a finite number – infinity just doesn’t work that way.<\/p>\n

Every choice you make defines your campaign’s constituent building blocks more clearly, excluding still more of the possibilities of character and plot. But always, the number remaining is still<\/em> infinite.<\/p>\n

In practice, it can be argued, none of that matters. A whole bunch of those remaining possibilities will be virtually indistinguishable from each other. Use one of them for an adventure and you exclude not only that plot, but all those variations that are too similar to it.<\/p>\n

In other words, you can, in this case,<\/em> subdivide infinity into some finite number of categories, each of which will be infinitely large, but each of which has common elements that will distinguish one infinity from its neighbor.<\/p>\n

If your category definitions are too broad, you violate the axiom that makes them useful – because you can<\/em> run more than one adventure from the ‘category’ and have them be sufficiently different from each other to be acceptable.<\/p>\n

The Lesson Of Infinity for GMs<\/h3>\n

That matters, because a lot of people don’t come up with an adventure and then classify it within their taxonomy; they pick a broader category that has not been represented for a while within the campaign (or at all), choose a sub-category that sounds interesting, and use the description as a starting point, a template,<\/em> for the design of the adventure: “I think I’ll do a heist plotline next, that sounds like fun…”<\/p>\n

It’s all about the way you think about the different types of adventure that you can run. Too narrow<\/em> a definition yields adventures that are potentially too similar to each other being permitted – boring! – while too broad<\/em> a classification structure fails to isolate the discrete combinations that will be of greatest interest. Neither<\/em> is particularly helpful.<\/p>\n

So, the next time you’re creating something new – be it a campaign, or an adventure, or an encounter, or an NPC – pause for a moment to review your mental taxonomy. Is your system of thought too narrow or too broad? Are you, in fact, making it harder for yourself?<\/p>\n

Infinities demand respect. Anything less, and the finite eventually breaks down under the burden, like a campaign that’s run out of ideas.<\/p>\n

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This image combines “sunrise-1756274” by pixabay.com\/qimono, “stars-1246590” by pixabay.com\/Free-Photos, “fireworks-1885571” by pixabay.com\/nosheep, and text rendered using cooltext.com, with compositing and additional editing by Mike.<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"

New years are about new beginnings, about punctuating the year that was, to separate it (however artificially, however optimistically) from the year that is to come. This article is about new beginnings, and being aware of the options you have, but it might not seem so, at first. Bear with me…. One of the hardest […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false},"version":2}},"categories":[65,67,70,74,91],"tags":[237,109,116],"series":[],"class_list":["post-23185","post","type-post","status-publish","format-standard","hentry","category-campaign-creation","category-dnd","category-gm-ing","category-mike","category-plans-and-prep","tag-adventure-creation","tag-dm-advice","tag-game-mastery"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p1toiD-61X","jetpack_sharing_enabled":true,"jetpack-related-posts":[],"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.campaignmastery.com\/blog\/wp-json\/wp\/v2\/posts\/23185"}],"collection":[{"href":"https:\/\/www.campaignmastery.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.campaignmastery.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.campaignmastery.com\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.campaignmastery.com\/blog\/wp-json\/wp\/v2\/comments?post=23185"}],"version-history":[{"count":8,"href":"https:\/\/www.campaignmastery.com\/blog\/wp-json\/wp\/v2\/posts\/23185\/revisions"}],"predecessor-version":[{"id":23200,"href":"https:\/\/www.campaignmastery.com\/blog\/wp-json\/wp\/v2\/posts\/23185\/revisions\/23200"}],"wp:attachment":[{"href":"https:\/\/www.campaignmastery.com\/blog\/wp-json\/wp\/v2\/media?parent=23185"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.campaignmastery.com\/blog\/wp-json\/wp\/v2\/categories?post=23185"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.campaignmastery.com\/blog\/wp-json\/wp\/v2\/tags?post=23185"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.campaignmastery.com\/blog\/wp-json\/wp\/v2\/series?post=23185"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}