Will's Blog: Permission or Forgiveness?

[Forestidia86]

Something like a philosophical wiki is the Stanford Encyclopedia of Philosophy: Entry for Philosophy of Maths
There are more entries for more specific subjects like Intuitionism in the Philosophy of Mathematics etc.

This thread reminded me somehow of the Zeno's Paradoxes.

[WillLem]

I'm happy to learn more here. I merely didn't see any examples and don't have much experience here myself. I hope that explains my math-slanted reply.

I'm equally interested in both sides: mathematical and philosophical. I regret the way this conversation started really.

It still sounds to me like you want to understand the uncountability of the reals. Your replies to other people also suggest interest in distinguishing countable and uncountable sets.

Yes, that's definitely a big part of it. I want to know how to get from 0 to 1 without missing a decimal, which is of course impossible. This idea both fascinates and confuses me.

Ok, I think you have a confusion between the concept of a sequence vs a set.  A sequence is ordered and countable.  A set is not ordered and doesn't have to be countable.

I think I understand this now; Proxima has also explained it the same way.

As already explained by others, it is actually impossible to list out all the real numbers, or even just the ones between 0 and 1, in any kind of lists.

In that sense, it seems like every number is a potential infinity in itself, given that decimal places can theoretically go on forever.

surely you can agree that human intuition isn't infallible either?

Absolutely: this thread is a fine example of my own human tendency to not be infallible!

No, but your initial attempts, at least in their wordings, seem to show you haven't even bothered questioning your own understanding of those findings.  Shouldn't one first "question and evaluate" one's own understanding of the things one is trying to challenge, lest the attempt only wound up challenging a completely distorted version not congruent to what the findings were actually saying?

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That's never how things work in any academic disciplines.  Right from the beginning as the initial version of the theory or framework is still being formulated and explored, things are already questioned, discussed and critiqued, and things get revised multiple times in the journey towards the formulation you see today.

I think there's been a very unfortunate misunderstanding in all of this, and that's the notion that I somehow reject academic rigour.

I absolutely do not: I have a Master's degree in Music, and wouldn't have got it without embracing the academic side of that particular discipline.

Similarly, I have nothing but the utmost respect for mathematicians, and Mathematics as an academic discipline. I regret the way I have approached this topic because mathematical logic, infinity and other such concepts are things I've always been fascinated by; I really don't know what made me think that coming at it with a seemingly anti-maths/anti-discipline angle would be a good idea.

I guess that it's a side of the argument that I felt needed some exposure for whatever reason, and it was somewhat spontaneous on my part. As ccexplore has said: I should have examined my own understanding of that side of the argument before posting about it.

Anyway, what's done is done. I appreciate the time that people are taking to reply to my posts and explain the various points and help to fill the gaps in my knowledge and understanding. Having settled from the misguided thoughtgasm that was my original post, I can now see the subject with a lot more informed clarity.

I'm lucky, really, that this forum attracts a lot of clearly very educated people with whom this discussion is possible at such a depth.

This thread reminded me somehow of the Zeno's Paradoxes.

Zeno's Paradoxes are great fun to mentally wrestle with every now and again. In fact, it was these that first introduced me to the possible confusions surrounding the concept of infinity in the first place.

[Simon]

Instead of example members plus ellipses, to be rigorous you would just provide a formula that says you start with 1, and then you can keep creating the next integer by adding 1 to the previous.  You don't need any ellipses and there is no ambiguity.

Right, it's important that one may define ℕ without resorting to "...". Only once everybody agrees what this set should be and that it exists, we can write ℕ = {0, 1, 2, 3, ...} as shorthand or as a reminder, not as a definition.

In the beginning, it's acceptable to "just believe" that ℕ and ℝ exist, and later replace them with more rigorous definitions.

When one wants a completely rigorous definition of ℕ within ZFC, the most popular system of axioms for set theory, it gets elaborate: It's common to define ℕ as the smallest infinite von-Neumann ordinal. This happens to be the set of all finite von-Neumann ordinals, which we then interpret as natural numbers. But that requires an understanding of ordinals first, and why a definition as "the smallest ordinal that satisfies X" is sound.

I can understand 3 oranges belonging to a set. However, there is no basket that can be made that is big enough for (every-positive-integer) oranges, so how can every positive integer belong to a "set", as we understand it?

Does a train suffice instead of a basket? But more seriously:

Naively, a set X is a mathematical object such that, given any mathematical object y, the statement y ∊ X "makes sense", i.e., it is either true or false. Also, sets may not be "too big" such that one runs into Russel's paradox or similar problems.

Thus, the set ℕ of all natural numbers exists; you can tell me with a straight face that 3, 5, and 329 belong to ℕ (because they are natural numbers), and you can tell me that ♥ and M aren't natural numbers, thus don't belong to ℕ.

As long as one accepts the existence of infinite mathematical objects, one can feel reasonably safe that the existence of ℕ doesn't yet trigger any paradoxes or contradictions: Decades of work haven't found any. We can't prove that it's really paradox-free, but the fundamental obstacle is not infinite sets, it is that for any sufficently rich system of axioms, one cannot prove from itself that it's free of contradictions. So this is really the best that we can get.



If one is not satisfied with naive definitions and wants something more rigorous and founded on classical logic, ZFC is the most popular formal system of set-theoretic axioms.

In ZFC, everything that exists is a set. The axioms force that some sets exist, such as the empty set ∅, an infinite set (doesn't matter which), the two-element set for any two given elements, and some more things. Note that the existence of infinite sets is explicitly forced in ZFC, it's part of the design of this system of axioms.

Because everything is a set, but we still want to do mathematics similarly to how we're used to, we'll model our desired mathematical things using sets. E.g., we define the natural number 0 to be the empty set ∅, the natural number 1 is the 1-element set {∅}, the number 2 is {∅, {∅}}, an ordered pair of two things x and y is the set {x, {x, y}}, and a function, a.k.a. a mapping, is a set of such ordered pairs.

It's similar to how everything in a computer (text, numbers, images, sound) is only a sequence of bytes under the hood. It's rarely necessary to consider the bytes: We don't talk about bytes much at all, we talk about numbers and texts and images, and it all makes sense to both of us. But if the need arises, we know how to look under the abstraction and examine the raw bytes.



Infinite sets are exactly those sets X that admit injections X → X that miss elements. (An injection is a function that never takes the same value at two different inputs.) For example, ℕ is infinite because the function n ↦ n + 3 takes only different values for different inputs n and misses (never takes as value) the first three natural numbers 0, 1, 2.

Finite sets are exactly those such that their size is expressible using a natural number.

It's not obvious that every set satisfies exactly one of these two, but it's provable in ZFC.

The nature of infinity as a philosophical or artistic idea will likely be fundamentally different to such a property of a set to admit certain functions to itself.



If one rejects the readily-existing infinite sets, one can still consider ℕ as building instructions to produce ever more mathematical objects, each itself finite, beginning with 0, and call them numbers. This is the basic idea of Finitism.

When one dives into formal logic, one will eventually separate two languages:

• In the finitistic metalanguage, we conduct proofs. Each proof can only have a finite number of steps and argue about a finite number of symbols. (It doesn't matter that, inside the theory, that symbol means something infinite. The proof treats it as one thing that has properties, e.g., being infinite, whatever that may mean in the theory.)
  
• In the domain-specific language that only makes sense when talking about objects of the theory, such as ZFC, we can use the term "infinite" even though that makes no sense in the metalanguage.
  

E.g., we can argue about the magic in the Harry Potter books, and our argument will use magic-related words that have meaning within that theory, even though nothing magic-related makes sense in our outside world.

It's useful to keep the metalanguage as weak as possible, to avoid introduction of inconsistencies. Reason is again: We can't prove that the metalanguage is consistent merely by using the metalanguage.

It's similar to how, in software, we don't want features unless there are good reasons to have them, as every feature has the potential to introduce bugs.

I have a Master's degree in Music

Hats off, then. It's one of the hardest subjects to even get admitted.

Regarding music, the "most infinite" thing that comes to my mind is the Shepard glissando. But maybe you have even better examples.

-- Simon

[ccexplore]

I have a Master's degree in Music

Hats off, then. It's one of the hardest subjects to even get admitted.

Interesting.  I don't have any experience to say anything about this one way or another; any guesses as to why it seems to be so difficult to get admitted?  Are they just pickier?

I tend to think of law and medicine as common examples of degrees that are often considered difficult to get into and just as difficult to finish.

[Simon]

I have a Master's degree in Music

Hats off, then. It's one of the hardest subjects to even get admitted.

why it seems to be so difficult to get admitted?  Are they just pickier?

Likely, it varies by university. Still, music and sports entrance exams are highly skill-based. I think it's common that the student needs to play two different musical instruments really well, and pass other skill tests about for musical intuition.

Law and medicine require excellent grades in high school, but I don't think they commonly have exams for entry otherwise. skill. At least from personal experience, good grades are easier to get than getting good at several musical instruments.

Medicine in Germany occasionally requires the best possible high school grade, 1.0 after ceil-rounding to tenths. If you don't make this, you can still get in, but you're placed in a queue first. Each year, a fixed ratio of slots are filled from the queue's front, and another fixed (small) ratio are filled from the queue at random. It may take 5 to 10 years to be admitted for medicine via queue. But at least you can get in even with decent grades if becoming physician is your major goal in life.

-- Simon

[kaywhyn]

It can certainly depend on the school, as some schools have a more rigorous curriculum than others and therefore will be harder, but I think the difficulty of your major will be about the same regardless of which college/university you attend. My alma mater UCLA is highly prestigious and has in recent years finally beaten out the oldest UC school, UC Berkeley, out of the top spot on the list of top public schools in U.S News and Times Report, when we been 2nd place on the list behind Berkeley for many years. At UCLA, the graduation rate for math majors is only 2/10, i.e, 20%, and so I was one of the few 2/10 who graduated with a math degree. Math is definitely a hard major, moreso at UCLA, but I'm sure if I had studied math at another university in California it would still be up there high in difficulty, although maybe not as much. It depends on the school. Besides UC's, another major school system are the CSU's (California State University). CSU's aren't as rigorous as the UC schools, but they're still widely regarded as some of the best schools in California. In particular, CSUF and CSULB are one of the most prestigious in this system. The former was where I attended grad school and did my Master's in math after I graduated from my undergrad at UCLA, while the latter was where I did the teaching credential program to get my teaching credential in math after I graduated with my master's from CSUF. When I did my master's in math, it was nowhere near as difficult, but I still didn't consider it a complete pushover, as I still had to study in order to do well in my math classes there, although I didn't study anywhere near as much at CSUF as I did at UCLA. At the same time, I think master's degrees in the same major as the bachelor's are easier in general. In particular, all of my math classes at CSUF were pretty much all review of the math I studied at UCLA, hence played a big part of why my Master's was easier.

Regarding your Master's in music, WillLem, that's awesome and congrats. I've loved music a lot as a kid, and I still do. I consider the arts essential to have in school, so it would disappoint me greatly if music and other electives, like band, which I didn't take, were taken away from schools. I took choir every single year in middle and high school, so yes, ladies and gentlemen, I love to sing, and I know the basics of music theory (how many beats each note is, the letter names of notes on the staff, etc). I do not know how to play any instrument, although I did take a basic piano course during summer school one year back when I was still at UCLA for my undergrad. I've always wanted to learn how to play the piano, though. Nowadays, I listen to radio music a lot, so I always have music playing in my car and while I'm working at my desk (I have an iHome). Perhaps I should consider finding a Zoom course that people are doing during quarantine from all this COVID madness. Or perhaps I should get back into recording singing videos, something which I only did for a few songs and for only a few weeks in college before I stopped completely. 

[ccexplore]

I guess things vary from country to country.  In the US, law and medicine are generally graduate/post-graduate programs to be entered after you got your bachelor's (which doesn't necessarily have to be from the same school), and do have entrance exams.  I didn't consider in other places they may start you off at your undergraduate studies and include those 4 or so years (plus the BA) as part of the program.

So I guess for music, what we're discussing would basically be about getting admitted to a conservatory, which I imagine is pretty competitive too.

To be clear, when considering difficulty of getting admitted, I'm mainly considering within the pool of applicants, as opposed to something like the average theoretical difficulty measured over any random person.  So yes, for the many people who have no interest or little aptitude in music to begin with, it would be hard if they were to apply, but the people who actually applied are more likely to already be both more interested and more talented than the average person.  Even so, within the applicants I can still see it being quite competitive to successfully get admitted.

[kaywhyn]

In general, yes, many grad programs are ones that you start right after you finish your bachelor's, and it's not at all an unusual situation to do your bachelor's at one school while doing grad school at a different one. In my case, I didn't fare so well academically at the university I did my undergrad, and so I would not had been able to get into any master's program at my alma mater. Now that I've done both a master's and a credential program elsewhere at two different schools and did well with both, I should be able to come back and get accepted into some kind of graduate program at my original alma mater. That's what I'm planning to do sometime in the future, but I'm not exactly sure when.

There are some post-baccalaureate programs that you can do while you're still working on your undergrad, but there's not that many. A good example would be working on a teaching credential while doing your bachelor's. Depending on the school, it is also possible to work on a master's AND teaching credential at the same time (they call it a joint master's/credential program). Where I'm from, the CSU's don't offer such a program, but the UC's and some other non-CSU's do. In the CSU's, you can only work on your master's in education after you finish and obtain your teaching credential. So, if I wanted to do a joint master's/credential program, I would had needed to go to another school to do it. While killing two birds with one stone would had been nice with a joint program and get both a master's and a credential at the same time, it would had meant way more work than doing both programs separately. Personally, I would had gotten overwhelmed by the huge amount of schoolwork if I had done a joint program, so it's all good.

I think it shouldn't be difficult to be admitted even if you're not already into the subject matter, and if you do get accepted, it's generally difficult to do well in a program where one isn't already well-versed and not as enthusiastic and interested about the subject. Of course, requirements are different for every graduate program, but generally one has to have a solid undergrad GPA in order to be admitted into any master's program. It's still possible to get admitted even if you don't, but usually you'll have to appeal. It's generally not required to have a bachelor's related to the subject of the program you're doing for your master's in order to be admitted, at least here in the USA. It might be different abroad. Like you mentioned, it varies from place to place. Usually it's the job that you're applying for that requires that you did your degree related to the work required. This is especially true for teaching beyond the high school level, although I have several co-workers who did their bachelor's in a completely unrelated field to the subject they're teaching. Some of my math teacher friends were history majors or something non-math related, for example. At the same time, teachers come into the teaching field in different ways. Some have worked on their credential almost right away after graduating from their undergrad, while others have done work in other fields for years before making the decision to go into teaching. Like with doing your bachelor's and master's at different universities, it's also not uncommon to do a program in one field for your bachelor's but then do your master's in a completely different field than your bachelor's. I did both of mine in math, as I feel like I can't see myself doing anything else besides math, maybe other than engineering or possibly something like NASA, as I do love science as well, astronomy in particular. It hasn't been done yet, but I'm quite certain that I won't be doing math for my PhD like I did for my bachelor's, master's, and teaching credential. Instead, I plan to either do a 2nd master's in education, which was originally my plan after graduating from my undergrad, or a PhD in a TBD field.

[WillLem]

Regarding your Master's in music, WillLem, that's awesome and congrats. I've loved music a lot as a kid, and I still do. I consider the arts essential to have in school
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perhaps I should get back into recording singing videos, something which I only did for a few songs and for only a few weeks in college before I stopped completely.

Thanks kaywhyn, congratulations to you also on your MA in Mathematics!

Music is a great thing to have in schools, even if it's only extracurriculur; quite a lot of British schools are dropping their music departments: it's usually the first thing to go when a school is struggling financially, and some schools are way more focused on the sciences so their Music departments are tiny and woefully underfunded.

Thankfully, private and independent music schools seem to be booming in the UK at the moment. I work for one, and it's a brilliant organisation. I hope to see even more of these as time goes on because they create great opportunities for young people who'd otherwise miss out on a formal musical education.

Coming back to the flat-Earthers thing for a moment: I have the ultimate proof that the Earth is not flat:

If it was, cats would have pushed everything off it by now!

[WillLem]

I'm almost worried about doing this again, but... well, in the name of freedom of expression, here goes...

Planet Earth might be expanding.

Arguments for:

• The continents fit together seamlessly on a smaller globe (this has been repeatedly proven and demonstrated by some of the world's finest scientists throughout history).
• Planets are known to vary in size; the smallest being composed of rock, the largest being composed of rock surrounded by layer upon layer of gas (however, the exact reasons/causes for this are still under continuing investigation).
• Planets have layers. Things that have layers tend to grow (onions, trees, symphonies, Lemmings levels ).
• There is no satisfying explanation for why continents and oceans are massive. (This is my own thought added to this - I haven't yet read anything or seen anything that has really satisfied the question of why there are such equally enormous land masses and oceans, with such distinctive shapes. Emphasis on the word yet; I'm an open minded guy).

Arguments against:

• There is nothing to adequately/conclusively explain the force behind, or cause of, this growth (but then - what about explaining the growth/variance in size of any planet?)
• There is nothing to adequately/conclusively explain where all the water came from (but then - why is Earth the only observable planet with water? We know that gas comes from liquids, which comes from solids. This simple, provable fact could account for it in some small way - maybe Earth is entering the "liquid" stage of its development, and will next start to become a gas giant...).
• Plate Tectonics Theory is grounded in years upon years of scientific discovery, thorough research from multiple branches of study, and physical exploration (OK, but humans have historically believed all manner of Theories with similar "certainty", and with similarly irrefutable evidence and conviction, only to later be corrected by another Theory).

You're all awesome.

Discuss!

[namida]

Again, this is the kind of thing that scientists would likely know - or at least suspect - if there was much chance of it. It's not the sort of thing they'd overlook.

For an example of a very similar concept being considered - look into the scientific discussions on whether the universe may be expanding.

[ccexplore]

A quick perusal of Wikipedia yields this article: https://en.wikipedia.org/wiki/Expanding_Earth.  Basically the theory had been proposed back in the 19th century, but ultimately had not been found to line up with the current physical evidence, so it is currently rejected by the scientific community.  To quote the article:

Examinations of data from the Paleozoic and Earth's moment of inertia suggest that there has been no significant change of Earth's radius in the last 620 million years.

Technically speaking, the earth probably experiences microscopic amounts of expansions and contractions simply due to factors like being at slightly different distances from the sun as it orbits the sun.  The resulting small temperature and gravity changes will have some effect, though that's probably not what you are picturing in mind.

The earth's internal temperature is expected to cool down over time, because heat radiates outwards and the internal sources of heat won't last forever.  The heat from the sun will not make up for this (I think the sun's heat mostly just drives the weather and the carbon cycle).  How the internal cooling affects the size of the earth I'm not sure, though usually solids contract as they cool.  Again, we're talking very tiny amounts here over very long stretches of time.

From a quick skim of the literature on Wikipedia, the prevailing plate tectonics theory explains that the continental plates movements over time can, from time to time, bring all the continents together into a supercontinent.  Such a movement would then force the colliding plate boundaries to fit against each other by force, and then later as the plate movements break them back apart again, you would see the separated continents with matching boundaries.  So instead of starting off unbroken, the matching boundaries actually occurred during the times when plates were pushed against each other.  The idea of an expanding earth breaking apart a mono-plate to form continents is a valid hypothesis, but not necessary to explain the matching continental boundaries, and apparently not supported by other current physical evidence.

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There is nothing to adequately/conclusively explain where all the water came from (but then - why is Earth the only observable planet with water?

Not really sure what this have to do with the earth expanding or not expanding?  Anyway, one thing to keep in mind is the idea of a habitable zone.  Liquid water can only exists under fairly narrow ranges of temperature and atmospheric pressure.  Otherwise they are either locked away as ice, or has long since evaporated/sublimated as vapor.

You can browse Wikipedia for details, but I do believe most other locations in the solar system have some form of H20 (or at least are believed to have them--direct evidence would require sending out spacecrafts to explore, so it will take time to get good coverage of observations), just typically not in liquid form.   Comets are mostly ice and dirt if I recall correctly.  Some moons of the outer planets is theorized to have water deeper inside.  The cores of the gas giants may have a layer of ice, etc.  I don't even know for sure whether earth actually holds the most amounts of H20 (of any form) in our solar system, but even if it does, it could just reflect the planet being in the habitable zone where liquid water is stable.

maybe Earth is entering the "liquid" stage of its development, and will next start to become a gas giant...).

That's not expected to happen.  The earth start off hotter than today and will cool over time.  This is not conducive to turning things into gas.  Moreover, my understanding is that the general elemental makeups of the inner planets are too different compared to the outer gas giants to credibly be able to go from one to the other.

[WillLem]

The earth start off hotter than today and will cool over time.  This is not conducive to turning things into gas.

I read about this last night - apparently, there has been evidence to suggest that Earth started as a gas giant and that it all happened the other way around.

Coupling this with the hypothesis that the habitable zone may be something that could itself expand, this could mean that Jupiter and Saturn may one day be habitable!

[ccexplore]

Coupling this with the hypothesis that the habitable zone may be something that could itself expand, this could mean that Jupiter and Saturn may one day be habitable!

Yeah, I wouldn't necessarily celebrate that even if it were to happen.  Siberia and Death Valley are technically "habitable" but not really the kind of climate you as a human would want to live in year-round.  Significant changes to the current habitable zone will very likely push Earth towards either end, if not put it entirely out of the zone altogether.  Remember that there is vast distance between Earth and Jupiter--you are crossing both Mars and the asteroid belt.  A change large enough to put the outer gas giants into habitable zone will almost certainly be catastrophic to Earth.

From what I remember, our sun is not expecting to have significant changes for 4 or so billion years.  Eventually at some point, as it started running out of hydrogen on the surface, it will turn into a red giant and expand greatly in size.  That will change the habitable zone for sure--but we're also talking about an expansion of the sun so catastrophic that all the inner planets including Earth, are expected to be swallowed up by the expansion.  Before all that happens I don't believe the sun is expected to have a significant enough change in its output to change the habitable zone, not to the degree of the zone boundaries crossing one let alone multiple planetary orbits.

[mobius]

I don't have time to read through all this so I apologize if this was already covered but I don't think it was;

On Math:
Do you think numbers exist? Or to put it under another way which of the "three schools of thought" do you prefer? According to numberphile that is https://www.youtube.com/watch?v=1EGDCh75SpQ This video explains the three theories better than I can.

Platonism; Numbers are real; all mathematical claims are true; they exist; they're abstract objects.

Nominalism; Numbers are not "real" objects; they just describe things that exist. Math is only true claims about the world.

Factionalism; Numbers do not exist.

I'm torn between nominalism and factionalism. I'm not sure as it may depend on the definition which could get a bit squirrelly.

The first two fall short especially when pushed to limits. E.g. start talking about the more abstract and complex parts of math like infinity or imaginary numbers.
If numbers are real; that is some kind of objects that exist in the world; where are they, what are they?
If numbers are merely a language to describe reality what is the reality that is being described with a number like pi? It's just an approximation

Deep down I believe mathematical claims are 'true' merely because we say they are and we decide what truth is in the first place. We created this complex theory in our mind and it ends at our mind. I believe all concepts are 'fictitious' in that sense or illusory. And I don't mean we 'create' it in the same sense that you might create a song or work of art. It is created when you look at a mathematical problem. There is only one thing that can happen; one way it can work because you have the brain that you have that works in this way. Math is an aspect of our mind; a language, a code that describes the world or makes up the world. Perhaps it is universal. Or it might turn out that other intelligent life operates with a totally different set of mental tools instead....

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On Earth might be expanding;
Honestly I don't find these arguments very compelling sorry. I've never heard this but it doesn't make a lot of sense to me.
-I thought this was why the Pangaea theory was formed? And I thought Pangaea is/was proven by plant/animal fossils being similar in certain places like Africa and South America etc.
-I don't really see what's strange about continents and oceans being massive. Earth imo is basically a 'water world'. It's 78 (or around that?) percent ocean.

They basically know (imo) how planets form and how/why they grow. In our case there was the 'period of large bombardment' when the solar system was young and there was debris (asteroids large and small) everywhere flying around. Larger objects have more gravity so they attract the most stuff and grow over time.
Nowadays (the past million(s) of years) there's relatively a tiny amount of asteroids and debris flying around, not nearly enough (that I'm aware) to make planets grow.


Are you familiar with project Kepler space telescope? Over the past ~10 years Kepler has discovered (with the help of the so-called citizen scientist project) over 1,000 exo-planets. These are planets (mostly gas-giants but many are more like Earth's size) around other stars. Kepler has only looked at a tiny portion of sky so far. Scientists now estimate there may be more than twice as many planets as stars in the observable sky (which means a huge huge amount).

What's interesting to me is they've discovered many "super earths". These are earth like planets but larger and often around smaller cooler, older stars. And what they've found so far suggests these might be more abundant than planets more like earth AND they might have calmer more uniform temperature and weather and geothermal activity. And these stars live much longer than our sun will as well (as long as trillions of years...) Meaning life would have longer to get a start; and easier time surviving and longer to live and evolve.
In hubris we always say "Earth is perfect for life" But it may turn out we our unlucky and there are in fact BETTER places for life elsewhere.

Conversely with fewer upheavals and difficulties these planets may have a HARDER time producing life; as maybe evolution needs those challenges to adapt and produce more complex life. 

anyhoo It's a very exciting time for science!

What are you're thoughts on alien life?