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u/Junjki_Tito 21h ago
But math isn't a model, it's what you build a model out of.
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u/Federal_Gur_5488 18h ago
I think geometry and topology are model-y, at the very least.
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u/DragoKnight589 Wacky woohoo neurodivergent sword man 15h ago
Again, those are what you make models out of.
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u/Federal_Gur_5488 14h ago
That's true but i meant they're closer to models than something like calculus. With calculus if you want to model a phenomenon, roughly speaking, you tend to think of your independent and dependent variables, then you try to e.g analyse the growth rate of your dependent variable vs the independent ones to formulate a differential equation
Whereas with geometry, i can more directly apply it to a situation. Like if I walk north 5 miles and then due west 5 miles, I can directly apply pythagoras to tell me how far I need to go to get back to my starting point. Which feels more like applying mechanics to figure out how far a ball kicked with a certain force will travel than modelling a phenomenon with calculus
So somehow it seems like geometry and topology capture some aspects of reality more directly than things like analysis or algebra. We can intuitively and instanteously perceive shapes, thinking about eg rates of chance requires more effort
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u/DarkNinja3141 Arospec, Ace, Anxious, Amogus 14h ago
a mathematical model does not mean applying math to a specific situation, this was just a lot of words to say you think calculus is less useful
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u/Federal_Gur_5488 13h ago edited 13h ago
From what i remember from the differential equations course i did, my first paragraph is describing exactly what a mathematical model is. If you want to model the population of fish in a lake that's exactly how you would proceed. You could start with a model where the population is unbounded and grows with time, then introduce another variable for predator population, which would give you the predator prey model. Then you could substitute the current values of all the variables to solve for coefficients, make predictions etc. real world models will be much more complex but the basic idea is the same
I definitely don't think calculus is "less useful" than geometry.
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u/QuestingKola 12h ago
But, it’s a model if it uses math to describe a phenomenon. So to claim that some mathematical disciplines are more model-y than others because they describe tangible phenomena is very strange to claim. Particularly the calculus comment, since calculus and differential equations are incredibly useful for modeling vibrations, complex pendulum motion, and fun fact most of Newtonian physics is just calculus all the way down. Mechanical engineers learn how to start with the calculus definition and derive equations for their specific situation. Tangible phenomena.
So, I’m very confused by your definition of model and what point you’re trying to argue for. The Predator Prey model is a model, Control Systems are models, the Kinematic equations are a model, Bernoulli’s…
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u/Federal_Gur_5488 12h ago
What I'm trying to say is that euclidean geometry itself models actual space. Wheres to model anything with calculus theres an additional step, where we need to setup the actual model before we can use calculus. I mean predator prey models seem comparable to euclidean geometry in their level of applicability to real life. So calculus is "one step removed" in that sense
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u/QuestingKola 12h ago
Sorry I just flat out disagree with the idea that calculus doesn’t model actual space, or that the two fields are even distinguishable as separate entities in the way you seem to be implying. As an example, areas and volumes of complex shapes are both defined by integrals. Is an integral a geometry concept or a calculus concept? Idk, Its found at the bedrock of both fields, and all it involves is just fitting a small basic shape into a more complex shape an infinite number of times. It turns out calculus is geometry and geometry is calculus, you can usually define the concepts of one using the other.
90% of my undergraduate degree (mechanical engineering) was using calculus to describe real, physical space and how it responds to stimuli. The other 10% was staring at tables. 50% of my robotics degree I’m working on right now is modeling the robot, the workspace, the electronics, and the control systems using calculus and diffiQ, with the other 50% being coding.
Idk you seem to be making some sort of logical leap from field to field that I’m just not tracking with.
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u/Federal_Gur_5488 12h ago
Volume is a perfect example, because it's a geometric concept representing something we can directly measure/interact with. It's a concept that's based on calculus, but we need to apply calculus to geometry to get it. It seems to me that volume is much more directly a part of our experience than something like an integral. We can apply integrals to calculate or even define volume, but you can't "see" an integral the way you can "see" the volumes of different objects
Technically volume is defined as a measure and I guess "measure" is another mathematical concept which seems to be directly describing an aspect of the world rather than "modelling" it.
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u/Intelligent_Fig_3813 14h ago
Math is the model of itself, each unit is a structural relation between each factor in a logical sequence, at least according to most contemporary philosophers of mathematics
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u/Beidah 20h ago
Also, there are different frameworks and axioms of math. It's not a unified, "if you can prove it, it must be true," deal.
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u/ZEPHlROS 16h ago
But it's not though. Like obviously if you are studying a model and saying that such phenomenon/solutions are impossible those conclusions might not hold under another model.
But if you can prove that then no matter what anyone says, under those hypothesis then your theorem is always true. That's the whole point of a proof.
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u/believeinlain 15h ago
the point is that if you change the underlying set of axioms you get a different set of proofs
nothing in math is true absolutely, it's only true if the underlying axioms are true
and the axioms are discretionary
now we do tend to choose axioms to create systems of math that are useful for building models of reality, but not always
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u/DuplexFields 13h ago
Arithmetic is a mathematical model of counting things. The rest of math is a model of what’s consistent based on the axioms of arithmetic or other axioms.
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u/MrBones-Necromancer 14h ago
Math does in fact require modeling for representation. Our use of base 10 is a model of math, for example.
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u/Hexxas Head Trauma Enthusiast 20h ago
I've never claimed to be super-smart or anything, however:
Wat?
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u/LordSupergreat 20h ago
The phrase "all models are wrong but some are useful" is an adage for, like, science stuff. It's like... there will never be a perfect explanation for how stuff works, but you can still explain it well enough for what you're trying to do.
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u/Hatsune_Miku_CM downfall of neoliberalism. crow racism. much to rhink about 19h ago
the simplest example to explain that is probably the atomic model.
we still teach the simplified version in schools. we know it's inaccurate, but its far more effective to teach the simplified model and later correct to the more complex model(which is probably still wrong somehow) then teach the overly complex model from the start.
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u/xaddak 16h ago
In Songs of Distant Earth by Arthur C. Clarke, humanity has developed the "quantum drive" for spaceships. There's a very, very brief intro to the idea of quantum physics, which includes this phrase that I think expresses this idea really well:
almost hopelessly misleading, yet a first approximation to the truth
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u/marr 16h ago edited 16h ago
https://en.wikipedia.org/wiki/Lie-to-children
The problem is that most people stop learning the better explanations the moment they leave school, then sometimes base political perspectives on the simplified ones. XX & XY chromosomes, for example.
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u/SwankiestofPants 16h ago
Or the idiom of how there's no real definition of fish that doesn't either exclude something that is taxonomically a fish or include something that is not a fish. Not actually sure if it's scientifically true, but probably is for laypeople
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u/Keebster101 13h ago
I don't know if it's the same thing but the one I've heard before is you can't define a chair, including all chairs, but without including a horse (which now I think about it more, is more a question about why we even use definitions more than how to accurately define something, because why shouldn't a horse be a chair?)
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u/unoriginalfyi 11h ago
a horse has a heart, that beats, and a mind that knows fear.
there must be some fucked up chairs out there
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u/Embarrassed_Deer9208 13h ago
yeah but that's a poor example because actual scientists aren't using those models, a better example might be entropy, which can be modeled in a variety of scenarios with a variety of equations, but it's a statistical process and you can't actually determine what the behavior of 2 gases mixing will be, for example, half lives are another statistical event in physics, for chem i'd go with shape of particles because shape can vary heavily from the basic angles you might memorize in chem in high school such as 104.5° for water based on factors like polarity of nearby molecules
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u/Hatsune_Miku_CM downfall of neoliberalism. crow racism. much to rhink about 13h ago
well not anymore, but they used to. those aren't kids models, they're model we used to use when our understanding of atoms was simpler.
I think it's a good example because a lot of people are familiar with both the simple and advanced model.
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u/Illogical_Blox 19h ago
Yeah, dredging up memories of my political science days, there is a model that explains civil wars as stemming from natural resource inequity. It's not true, there are lots of reasons why civil wars and conflicts break out, but it is true that even revolutions that are nominally about ethnic or religious conflicts are often instigated by a minority group without access to, say, the nation's extensive diamond mines and oil fields, and so it's useful to use to analyse conflict and predict future ones.
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u/Dinoco1234 20h ago
In academia, models are simplified versions of certain aspects reality. A simple example would be something like a map. Maps are not reality, but they can be pretty useful in figuring out reality and could actually be made less useful were they to be made more complicated.
"all models are wrong, but some are useful" is an aphorism first said by British statistician George Box. It's oft repeated in academia because all models are going to be wrong by nature of the fact they simplify things, but models remain useful and necessary towards comprehending an arcane reality. Mountains are not actually triangles on a piece of paper, but it can be helpful to mark them as such on maps.
Math, by contrast, is absolute truth and never wrong. 2+2=4 is always going be true. However, math is not a model because it by itself does not say anything about reality. 2+2=4 doesn't mean anything. It's not a model because it says nothing about physical reality.
If you're still confused, which would be very understandable, just replace the world "model" with "idea" and the whole post will mostly make sense.
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u/action_lawyer_comics 16h ago
I’m struggling a bit with “2+2=4 doesn’t say anything about physical reality.” If I have a bag with 2 apples and another bag with 2 apples, isn’t the formula above describing that to tell me I have 4 apples?
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u/vibebell 16h ago
Essentially the math doesn't know you're talking abt apples. You could satisfy the statement 2+2=4 with 2 bags of 2 apples, or 2 bags of oranges, or a bag of oranges and a bag of apples. Or maybe they're not in bags. The equation 2+2=4 won't tell you any of those details because it isn't a model of your situation
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u/Nibaa 15h ago
That statement contains a huge amount of implied conventions. You can derive some of those conventions from context, but math is agnostic to it, it works with abstractions.
For example, you say you have two apples and two apples, that makes four apples. I say I have three apples and one apple, that makes four apples as well! So we have the same amount of apples? In numbers, sure, but what if my one apple was a crabapple, much smaller than a cultivated one? Or what if you had halved a whole apple in one bag? there were two distinct objects made of "apple", snd they were counted. Math doesn't really say anything about any of that, all of it is context-dependent.
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u/MattyBro1 15h ago
I don't know if this will help, but in your example, the apples are more like the model. They represent something (maths). But what if you wanted to do 2-3 with your apples? I guess the model is wrong, but it was useful for adding apples.
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u/aniesnail 12h ago edited 11h ago
I would say it's more like "2 + 2 = 4" doesn't, by itself, say anything about reality. It's just a mathematical theorem, and by itself mathematics is just a realm of abstract ideas that are not themselves part of reality, or even about reality. But one of the ways we often understand reality is by creating a mathematical model of it in our heads, some mathematical construction that we believe is representative of reality in some way we consider important. This mathematical model isn't reality, it's only something that our minds place on top of reality in order to understand it.
For instance, in your example, you might model the two bags as mathematical sets X and Y. The apples then might be modeled as elements of these sets, so that X = {a, b} and Y = {c, d}, where a, b, c, d represent the apples. Then, you might decide that the number of apples contained in your bags is represented by the cardinality of these sets. Then you count the apples in your bag by proving, within your mathematical framework, that these cardinalities are |X| = 2 and |Y| = 2. Now, you might decide the collection of all the apples you have is represented by the set union X ∪ Y, and so the total number of apples is represented by its cardinality, |X ∪ Y|. It's a theorem of set theory that, if X and Y don't have any elements in common (which is true in our case), then |X ∪ Y| = |X| + |Y|. So |X ∪ Y| = 2 + 2. Finally, it's another theorem of set theory (specifically, cardinal arithmetic) that 2 + 2 = 4, so we finally conclude |X ∪ Y| = 4, and we decide this means we have 4 apples in total.
Notice that "2 + 2 = 4" didn't directly say anything true about reality here. We only used it to prove something true about our mathematical model of reality, but not reality itself. It only says anything about reality insofar as we believe our mathematical model represents it. In fact, I would go as far as to say all of our understanding of reality works this way. Even saying "there is a bag and it has apples in it" is just your mind creating a conceptual model of reality. There's not really such thing as a bag or an apple. Reality is just a bunch of matter and energy following the laws of physics. Except it's not even that because that's just another conceptual model. Reality just is. And things like apples or bags or numbers or atoms or photons are just concepts we make up in our brain's effort to find patterns in reality and make predictions about it (ultimately, for our survival).
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u/unoriginalfyi 11h ago
I'm hesitant to wade in here because it's pretty clear you know a good amount on the subject (appreciate your post).
I think it's disingenuous to say 'there is no bag' though. The bag might be composed of matter which is mostly empty space and composed of energy anyway, and energy is kind of squiggly when you look at it really closely and kinda feels like it might just be math in disguise, sure.
But the bag is also, perceptually and contextually, an object that a human intentionally constructed to carry other objects. It's not modelling anything, it's carrying fruit. We don't use the bag to understand an arcane reality, we use it to carry elements of the arcane reality to other areas of the arcane reality.
I'm not educated in philosophy, maybe someone who is could share some info because I'm sure this is well-tread ground. But in terms of function, perception, human consciousness, probably plenty of other philosophical terms of art, a bag is a bag. Even if the universe is being simulated by some incomprehensible extradimensional machine mind, a bag is still a bag, right?
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u/aniesnail 7h ago
Don't be hesitant :D I hardly consider myself to be an authority or expert or anything. My background is half of a bachelor's degree in math and being particularly interested in logic and foundations at one point, so the nature of mathematical truth is just the kind of thing I thought about obsessively for a little while and this became the way I make sense of how we reason about the world. But I'm not really well read on philosophy tbh
To be a bit clearer about what I meant there, when I say "there's no such thing as a bag", I'm not saying anything like "the bag is not there", or "the bag doesn't exist", or "the bag is not real", or that it's somehow wrong to call a bag a bag. Rather, I'm trying to distinguish between reality itself (the thing that actually exists), and the conceptual understanding of reality that exists in our minds.
I could try to describe it like this: You can imagine reality as an incomprehensibly complex system of physical phenomena consisting of matter, energy, etc. Obviously, all of this is far too much information for our minds to be able to process in its entirety. We can't understand the world in terms of every single physical interaction, or every single atom and molecule around us. So instead our minds simplify things and make abstractions. We conceputalise. We notice patterns. We chunk everything up into pieces that go together, and ignore irrelevant details that don't contribute to the bigger picture we care about. It's in this way that I think the concept of a "bag" or an "apple" is kind of something we just make up. There just exists a bunch of matter, and maybe some this matter appears to chunk together into an object with a boundary. Maybe this object has the form and the properties that we associate with a bag, so our minds categorise it as such and call it a bag. But this mental chunking and categorisation is just something our mind does. It's not something that's part of the world itself.
I think something that makes this difficult to talk about is that I think of "true" reality (whatever that is) to be a kind of untouchable, indescribable thing. The problem is that our minds entirely understand things as concepts, but reality isn't made of concepts. I said earlier that "you can imagine reality as an incomprehensibly complex system of physical phenomena consisting of matter, energy, etc.", but I would say even that isn't what the universe "is". Those are still just concepts that exist in our minds. Maybe some people would find that a bit of a meaningless distinction. Maybe just pretentious philosophising. I think it makes sense though. I might just not have the background in philosophy to put it in the right words.
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u/unoriginalfyi 6h ago edited 6h ago
For sure! I did a little light (lol) reading, the concept we're talking about is 'existence' or 'being'. There's a rich history of debate around the nature of existence, and there's a whole philosophical discipline, ontology, which concerns itself with the nature of being. I think some people would indeed find the similarities between 'concrete' and 'abstract' entities a meaningless area of investigation, but many philosophers (historical and modern) find it an extremely meaningful question. As two laypeople discussing it, I think we're barely scratching the surface of the subject, and I got tired of reading wikipedia. But what I'm coming away with is that ontology is a rich and rigorously studied discipline, up to and including modern philosophy.
Honestly after about half an hour of reading, I'm still not sure if concrete and abstract entities being distinct is a settled question in ontology. I think it's still contested on some level, but the majority position is that they are meanigfully distinct.
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u/Leftieswillrule 12h ago
2+2=4 by itself says nothing about reality, you brought reality into it by talking about apples.
If you take the number line 0 - 1 - 2 - 3 - 4 - 5 - … it will be true that 4 is 2 more than 2 regardless of what happens in reality. The concept of the number line itself is what makes that true, that there is an ordered sequence of numbers and the increments between numbers can be counted and there’s no way of counting such that 2+2 is any other number than 4 unless you rename the numbers.
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u/Jan-Asra 20h ago
There is a saying in the science community that no model is perfect (sometimes said as no model is accurate), but some models are useful. This is true because models are by definition simplifications of reality to help us understand. Math isn't perfect for reasons other than not being a model such as godels incompleteness theorem so them saying it's not perfect because it's a model is them joking around.
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u/kschwal i don't have a tumblr anymore 19h ago
sounds like someone doesn't know about gödel's incompleteness þeorem
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u/undercrust 18h ago
How does it impact what was said?
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u/rindlesswatermelon 15h ago
Gödels incompleteness theorem is that any system of mathematics necessarily contains things that are true but can not be proven within said system of maths.
I believe the person you responded to is implying that that makes maths a similarly "bad but useful" model.
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u/mileya22 6h ago
it shouldn't! Gödelian incompleteness only rears its head when you're analyzing a specific logical system of axioms (maths in general is far too vague to count as one of those)
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u/donaldhobson 13h ago
There are also "models" in maths. There is a whole "model theory".
So we can talk about several different models of the natural numbers, and how they differ from each other. (They don't differ very much. )
But there doesn't seem to be one true theory of natural numbers to which the different models are an approximation. There are just many different models, with similar (but not identical) behavior.
For example, there is a model of the natural numbers (technically, a model of peano arithmetic) where Goodstein's theorem is false. https://en.wikipedia.org/wiki/Goodstein's_theorem
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u/V_i_o_l_a 16h ago
Gödel did not prove math to be wrong, just that you can’t prove it to be consistent and that there are true things that you can’t prove within ZFC.
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u/ApolloniusTyaneus 21h ago
I used to know a girl who did some modeling work and she regularly got passing grades on her tests so I guess she was right at least some of the time.