4 ms·

> ... including the "novel" math solutions, all of which appear to just be "a composition of solutions humans have developed and documented elsewhere" upon dee

> ... including the "novel" math solutions, all of which appear to just be "a composition of solutions humans have developed and documented elsewhere" upon deeper inspection. But that is precisely what human mathematicians do, prove new theorems by combining ones proven earlier. I don't see any fundamental difference in functionality between human intellectual contributions vs performant ML ones (LLM or otherwise). Whenever we listen or read text we are also predicting the near future content. Just like LLM's we sometimes correctly predict the next token or word, and sometimes incorrectly. > The "something deeply wrong" part about AI, that even most technology enthusiasts evidently do not seem to grasp, is that it is still fundamentally a statistical model [...] Imagine someone could pause the universe with a remote control, scroll back in time a little, press play again, and ask a slightly different question, etc. In such a thought experiment one could also collect the probabilities for a specific human predicting a next word. Implicitly the brain also has a corresponding statistical model, regardless of the construction being visible or hidden. I.e. human intelligence is also fundamentally a statistical model, so the only thing that remains from your claim is that machines for some unmentioned reason don't possess any "real" intelligence or critical thought... Is it possible that our aversion is simply driven by educational systems collectively and deeply ingraining into populations the idea that intelligence deserves the high costs commanded. Well of course this justifies higher wages towards the higher leadership positions, etc. Now it turns out that intelligence can be dirt cheap. We discover that the fact that "intelligence must be costly so don't question the costs of leadership" was never fundamentally true, so the real anger is this discovery of mismatch between the old claims which served to explain how every society that claimed to order itself and fill positions accordingly with "naturally pre-ordained individuals". Now we are seeing robots exceed average workers, for effectively a grain of rice.

deleted 2mo ago
[deleted]
mik1998 2mo ago
New and interesting mathematics is done by inventing new definitions and fields, not just combining old theorems to prove new ones.
- DoctorOetker 2mo ago
  It's not that different, when a human proposes a better definition vis-a-vis a competing one for example, they would defend this by certain desiderata. Often a mathematician or physicist will use their intuition to speed up the naive brute force of candidate well formed formula variations so that the desired properties emerge, postulating the existence of an intersection on multiple desiderata can in itself be viewed as a novel conjecture, to be proven or disproved. A very basic (unimpressive) example for an example desideratum is regularity or compactness. the tau=2 * pi substitution does make a whole bunch of expressions more slightly more regular and compact. That is something objective and measurable on a system of theorems. There is no mathematician's moat vis-a-vis machine learning at a fundamental level. There can be artificially sustained moat, if AI powers limit the distribution of say cryptographic advance capable models, in jurisdictions outside such AI powers, but even that would be expected to be fleeting and temporary...
  - manwe150 2mo ago
    I think the parent meant it is more interesting to pose new problems than solve them. Posing a new conjecture along the path to solving something is a close cousin, but still seems more bounded than proposing something novel to prove—if only because proving that something novel is also actually interesting is subjective and thus difficult for a different reason.
  - tripzilch 2mo ago
    You're conflating a discussion about current LLM capabilities with your fantasies about nonexistent future AI. LLMs act nothing like this, and the small example you're giving is only a small part of the things that LLMs can't do.
    - calf 2mo ago
      Not PC but is ideologically high-handed and inappropriate to accuse the other side of failing to stick to your desired framing of a discussion. Computer science is about what LLMs fundamentally are. If you implicitly focus on "actually existing LLMs", and require others do this, then that is not computer science. That is politics.
ddj231 2mo ago
I see this line of reasoning quite a bit and it’s a strange one to me. The arguer reduces the sheer complexity of human intelligence and language by saying “we are just running statistical models in our brains” and by doing so makes the leap that Llms are intelligent. It’s an incredible simplification of the human person, who has a deep inner life, a soul, desires, and a will. I don’t think the aversion to llms as intelligent has to do with the economics of paying intelligent agents more. I’d argue that it’s more fundamental than that. Humans are incredibly complex, and the world of sharing invisible things called knowledge, and the intelligent persons consuming such things which has been going on for thousands of years is far more rich than these synthetic outputs. When it comes down to it the ai has no inner life, its is dead. A useful coding tool sure. But I wouldn’t call it intelligent. One side example is just how bad these llms are at artistry. Just saying whatever should statically come next is not good art—and the outputs show it.
- alchemism 2mo ago
  I tend to think of it in reverse: not that LLMs are intelligent, but that humans are more empty than we…think we are.
  - jurgenburgen 2mo ago
    I think that reveals your cynicism more than it provides any useful insight about intelligence.
    - alexalx666 2mo ago
      No, it reveals that he is older than you. Not every insight needs to be useful.
      - jurgenburgen 2mo ago
        > Not every insight needs to be useful. Don’t they still need to be correct to be an insight? I don’t share his cynical opinion that “humans are more empty than we…think we are”.
digikar99 2mo ago
Probability is just one way to model uncertainty. While I understand the brain encodes uncertainty, I don't think probability is a good enough model of what it's doing. Secondly, if you think verifying a proof in mathematics, reasoning within (and not about) a formal system, or following the chain of a computer program that is already written is just doing token-based probabilistic predictions, I don't know what to say. Thirdly, machines don't have a notion of value or stake. There's no way for them to verify whether what they have produced aligns with your unstated values and preferences. We regularly do this with other humans. I don't give you (or even my parents or partners) the benefit of doubt regarding whether you know me better than I do. Sure, you might know some things about me, but it's ultimately up to me to verify if what they say is applicable to my current situation. It's really uncanny to see people develop this codependency with their chatbots. And corporates encouraging them to do so. I'm with you that intelligence is not something to be proud of. But I also think it is instrumental to understand the world. I'm still waiting for the time when an unconstrained-AI machine can live without reprogramming for an entire decade. We are still far from there.
- deleted 2mo ago
  [deleted]
- DoctorOetker 2mo ago
  > Probability is just one way to model uncertainty. I study physics, mathematics, probability, cryptography,... so forgive my skepticism: Show me how to model uncertainty without use of probability. Can you rephrase say diffusion, stochastic equations, quantum mechanics in this alternative framework? Can it at least make the same predictions? Or is it basically the same framework in parallel, just giving different names for each concept? Forgive my skepticism of such tall claims, and forgive my downscaling of anything else you say besides such a claim... > Secondly, if you think verifying a proof in mathematics, reasoning within (and not about) a formal system, or following the chain of a computer program that is already written is just doing token-based probabilistic predictions, I don't know what to say. I make no claims of the specific shape of the implicit model implemented by a certain human brain educated in a certain educational system. For example in English the implicit human tokenization might be presumed to lay relatively close to English syllables, while in Asian languages it might be "sub" strokes of characters etc. Such implicit tokenization can never be proven to "match the one of humans" not because of human superiority, but because different humans use different tokenization methods. There is no "one human tokenization method", but it's clear as day there is an implicit one: everyone knows the experience of knowing a word, knowing its approximate group-wise meaning (ignoring that when you think of "an apple" and when I do, we typically imagine a slightly different apple) yet having the word feel strange or discover some older literal meaning when decomposing it or looking it up in an etymological dictionary. Suddenly one can become aware of a sensible meaning as a composition of subtoken concepts. A child may perfectly know what "television" means and only later learn more exact meanings of "tele" and "vision", and upon repeating the word may feel the word "television" has changed meaning. This clearly demonstrates "tokenization" effects in human language comprehension, not just across cultures, but also across individuals within a culture. > Thirdly, machines don't have a notion of value or stake. There's no way for them to verify whether what they have produced aligns with your unstated values and preferences. We regularly do this with other humans. I don't give you (or even my parents or partners) the benefit of doubt regarding whether you know me better than I do. Sure, you might know some things about me, but it's ultimately up to me to verify if what they say is applicable to my current situation. It's really uncanny to see people develop this codependency with their chatbots. And corporates encouraging them to do so. That's a lot of different concepts conflated into one bullet point, so I split it up: The notion of values and preferences. They clearly demonstrate the ability to take into account values and preferences, from training corpus, from RLHF, from system prompts, ... we can't simultaneously point at censorship aspects and pretend their effective values and preferences to be absent. The censorship aspects are clear as day, so these correspond to values and preferences. Just like radicalization among humans, this can be due to exposure to radicalized content (akin to corpus data), from indoctrination (akin to RLHF), from "set and setting" (they may pretend to be aligned with one set of norms and values when standing in line to buy their new smartphone, but then reveal alignment with a different set of norms and values when conversing in some "private" online echo chamber). I see no grand difference between humans and language models here. Awareness of values and preferences of a conversation partner. Allow me to widen it to "Awareness of values, preferences and prerequisites of a conversation partner". This move (and I see it every time when people try to defend superiority of humans vis-a-vis what machines could be made to achieve with current technology) is so far from the principal variation, I recommend you reconsider this one. I constantly see people claim say human teachers are necessarily better than LLM teachers, but upon closer inspection the "human advantage" just boils down to asymmetric privilege. A human teacher in a specific school has access to a lot more than a random chatbot as implemented today: they probably know which courses and even which textbooks their pupils saw the semester before, they know which teachers their pupils got their information from, perhaps they even know most of their pupils from teaching some preceding course materials to the same class of pupils. Current LLM's are crippled by design not to accumulate knowledge over conversations for both purposes of cybernetic control as well as cost efficiency: we know how to do "source aware training" (so that statistically it doesn't just absorb claims from the corpus, but also maintains epistemic traces of where it sourced these factoids from), its perfectly possible to continue training interleaved with conversation rounds so that it bakes the evolving conversation as read knowledge into its weights instead of into a context window. Nothing stops you from implementing this in local compute, it would probably be even more computationally efficient in a local inference setting since we can ditch the context window, the context is impressed into the weights continuously, it could thus take into account earlier conversations and estimate your knowledge gaps etc, or learn from you. When you wish to serve inference to millions of human users, you don't want to store millions of diverging LLM weights into expensive VRAM, they financially prefer a single large set of LLM weights, and then some user-specific context window, so the users don't freak out when they learn personal information a friend or stranger entered and an LLM service just leaks it into your conversation! It's not that we don't know how to implement it, and there are great advantages for local inference in doing this, its just not good for branding. Codependency with chatbots. I think everyone agrees codependent relationships aren't very healthy, regardless if it's with humans or machines. May I ask if you feel the same about prostheses and medicine? Conflicts of interest arising from corporate ownership of infrastructure (both training and inference). Yeah I think this point doesn't provide fruitful discussion if most of us agree on such matters already, we'd just be lamenting the same things, and agreeing with each other over and over here. > I'm still waiting for the time when an unconstrained-AI machine can live without reprogramming for an entire decade. We are still far from there. Apart from budget, nothing prevents you from doing this today, if you continuously bake in the fresh episodic memories into the weights (instead of a context window) regardless if its text, visual imagery, audio, proprioception or other sensory data.
squigz 2mo ago
> Just like LLM's we sometimes correctly predict the next token or word, and sometimes incorrectly. There is no "correct" next word when it comes to communicating with an actual human.
- DoctorOetker 2mo ago
  Depending on what this specific actual human says next, the prediction will be truly correct or not.
andrei_says_ 2mo ago
> But that is precisely what human mathematicians do, prove new theorems by combining ones proven earlier. I’d say “understanding and building upon ones proven earlier”
- DoctorOetker 2mo ago
  yeah just correct people with an infinitely recursive definition, that'll keep 'em busy
tripzilch 2mo ago
> But that is precisely what human mathematicians do, prove new theorems by combining ones proven earlier. This is only a very very small part of what human mathematicians actually do. This is just a lack of imagination and/or self awareness on your part.
- niek_pas 2mo ago
  There's gotta be nicer ways of saying that.
  - OrderlyTiamat 2mo ago
    It's also not true, mathematicians are already saying that current AI is changing the field of mathematical research profoundly.
    - cicko 2mo ago
      So did chalk and blackboards.
    - tripzilch 2mo ago
      None of them are saying "this machine is doing exactly what we do", though.
      - deleted 2mo ago
        [deleted]
      - DoctorOetker 2mo ago
        they don't say it about each other either, though
  - tripzilch 2mo ago
    There's gotta be nicer ways of saying "this is exactly what mathematicians do", it's insulting.
3dddad 2mo ago
" prove new theorems by combining ones proven earlier." So? Mathematics is literally intangible scaffolding. If you dont know / don't believe 5+5 = 10 You cannot solve x+5 = 10 This is all make-believe stuff and nature by itself doesn't care of its existence.
- DoctorOetker 2mo ago
  The phrase "God created the natural numbers, all else is the work of man" is a famous quote by the 19th-century German mathematician Leopold Kronecker. You can basically read it as: the moment one has axiomatized mathematics to the point it supports natural numbers, the rest implicitly follows. The natural numbers (positive integers) are closed for addition, multiplication, ... One can perfectly model the integers with a pair of naturals: < M, N > ~ (M-N) Now we can have any < M1, N1 > and subtract < M2, N2 > without needing the ability to subtract natural numbers: < M1 , N1 > - < M2, N2> ~ (M1-N1) - (M2 - N2) = < M1 + N2 , M2 + N1 > ~ (M1+N2) - (M1+N1) We can similarily define addition of such tuples, or test equivalence without access to subtraction of naturals: < M1, N1 > == < M2, N2 > <=> M1 + N2 == M2 + N1 ~ (M1-N1) == (M2-N2) <=> (M1+N2)=(M2+N1) we can also still multiply such tuples: < M1, N1 > x < M2, N2 > = < M1*M2+N1*N2, M1*N2+M2*N1> Similarily, even though these newly defined integers (which can be positive or negative) don't support division, the same trick can be used to make a new compound tuple of integers closed for division, by only using multiplications. Probability is a branch of mathematics (probability already exists embedded in mathematics implicitly, probability theory involves the addition of eliminable definitions, syntactic sugar. The patterns are already there, just less explicitly manifest. Mathematics is itself a branch of logic. Do you reject like all of logic, and if so, what would you like us to evaluate the sentences you write to? You want us to evaluate your expressions as "true" or as "false"?

deleted 2mo ago

[deleted]

mik1998 2mo ago

New and interesting mathematics is done by inventing new definitions and fields, not just combining old theorems to prove new ones.

DoctorOetker 2mo ago
It's not that different, when a human proposes a better definition vis-a-vis a competing one for example, they would defend this by certain desiderata. Often a mathematician or physicist will use their intuition to speed up the naive brute force of candidate well formed formula variations so that the desired properties emerge, postulating the existence of an intersection on multiple desiderata can in itself be viewed as a novel conjecture, to be proven or disproved. A very basic (unimpressive) example for an example desideratum is regularity or compactness. the tau=2 * pi substitution does make a whole bunch of expressions more slightly more regular and compact. That is something objective and measurable on a system of theorems. There is no mathematician's moat vis-a-vis machine learning at a fundamental level. There can be artificially sustained moat, if AI powers limit the distribution of say cryptographic advance capable models, in jurisdictions outside such AI powers, but even that would be expected to be fleeting and temporary...
- manwe150 2mo ago
  I think the parent meant it is more interesting to pose new problems than solve them. Posing a new conjecture along the path to solving something is a close cousin, but still seems more bounded than proposing something novel to prove—if only because proving that something novel is also actually interesting is subjective and thus difficult for a different reason.
- tripzilch 2mo ago
  You're conflating a discussion about current LLM capabilities with your fantasies about nonexistent future AI. LLMs act nothing like this, and the small example you're giving is only a small part of the things that LLMs can't do.
  - calf 2mo ago
    Not PC but is ideologically high-handed and inappropriate to accuse the other side of failing to stick to your desired framing of a discussion. Computer science is about what LLMs fundamentally are. If you implicitly focus on "actually existing LLMs", and require others do this, then that is not computer science. That is politics.

ddj231 2mo ago

I see this line of reasoning quite a bit and it’s a strange one to me. The arguer reduces the sheer complexity of human intelligence and language by saying “we are just running statistical models in our brains” and by doing so makes the leap that Llms are intelligent. It’s an incredible simplification of the human person, who has a deep inner life, a soul, desires, and a will. I don’t think the aversion to llms as intelligent has to do with the economics of paying intelligent agents more. I’d argue that it’s more fundamental than that. Humans are incredibly complex, and the world of sharing invisible things called knowledge, and the intelligent persons consuming such things which has been going on for thousands of years is far more rich than these synthetic outputs. When it comes down to it the ai has no inner life, its is dead. A useful coding tool sure. But I wouldn’t call it intelligent. One side example is just how bad these llms are at artistry. Just saying whatever should statically come next is not good art—and the outputs show it.

alchemism 2mo ago
I tend to think of it in reverse: not that LLMs are intelligent, but that humans are more empty than we…think we are.
- jurgenburgen 2mo ago
  I think that reveals your cynicism more than it provides any useful insight about intelligence.
  - alexalx666 2mo ago
    No, it reveals that he is older than you. Not every insight needs to be useful.
    - jurgenburgen 2mo ago
      > Not every insight needs to be useful. Don’t they still need to be correct to be an insight? I don’t share his cynical opinion that “humans are more empty than we…think we are”.

digikar99 2mo ago

Probability is just one way to model uncertainty. While I understand the brain encodes uncertainty, I don't think probability is a good enough model of what it's doing. Secondly, if you think verifying a proof in mathematics, reasoning within (and not about) a formal system, or following the chain of a computer program that is already written is just doing token-based probabilistic predictions, I don't know what to say. Thirdly, machines don't have a notion of value or stake. There's no way for them to verify whether what they have produced aligns with your unstated values and preferences. We regularly do this with other humans. I don't give you (or even my parents or partners) the benefit of doubt regarding whether you know me better than I do. Sure, you might know some things about me, but it's ultimately up to me to verify if what they say is applicable to my current situation. It's really uncanny to see people develop this codependency with their chatbots. And corporates encouraging them to do so. I'm with you that intelligence is not something to be proud of. But I also think it is instrumental to understand the world. I'm still waiting for the time when an unconstrained-AI machine can live without reprogramming for an entire decade. We are still far from there.

deleted 2mo ago
[deleted]
DoctorOetker 2mo ago
> Probability is just one way to model uncertainty. I study physics, mathematics, probability, cryptography,... so forgive my skepticism: Show me how to model uncertainty without use of probability. Can you rephrase say diffusion, stochastic equations, quantum mechanics in this alternative framework? Can it at least make the same predictions? Or is it basically the same framework in parallel, just giving different names for each concept? Forgive my skepticism of such tall claims, and forgive my downscaling of anything else you say besides such a claim... > Secondly, if you think verifying a proof in mathematics, reasoning within (and not about) a formal system, or following the chain of a computer program that is already written is just doing token-based probabilistic predictions, I don't know what to say. I make no claims of the specific shape of the implicit model implemented by a certain human brain educated in a certain educational system. For example in English the implicit human tokenization might be presumed to lay relatively close to English syllables, while in Asian languages it might be "sub" strokes of characters etc. Such implicit tokenization can never be proven to "match the one of humans" not because of human superiority, but because different humans use different tokenization methods. There is no "one human tokenization method", but it's clear as day there is an implicit one: everyone knows the experience of knowing a word, knowing its approximate group-wise meaning (ignoring that when you think of "an apple" and when I do, we typically imagine a slightly different apple) yet having the word feel strange or discover some older literal meaning when decomposing it or looking it up in an etymological dictionary. Suddenly one can become aware of a sensible meaning as a composition of subtoken concepts. A child may perfectly know what "television" means and only later learn more exact meanings of "tele" and "vision", and upon repeating the word may feel the word "television" has changed meaning. This clearly demonstrates "tokenization" effects in human language comprehension, not just across cultures, but also across individuals within a culture. > Thirdly, machines don't have a notion of value or stake. There's no way for them to verify whether what they have produced aligns with your unstated values and preferences. We regularly do this with other humans. I don't give you (or even my parents or partners) the benefit of doubt regarding whether you know me better than I do. Sure, you might know some things about me, but it's ultimately up to me to verify if what they say is applicable to my current situation. It's really uncanny to see people develop this codependency with their chatbots. And corporates encouraging them to do so. That's a lot of different concepts conflated into one bullet point, so I split it up: The notion of values and preferences. They clearly demonstrate the ability to take into account values and preferences, from training corpus, from RLHF, from system prompts, ... we can't simultaneously point at censorship aspects and pretend their effective values and preferences to be absent. The censorship aspects are clear as day, so these correspond to values and preferences. Just like radicalization among humans, this can be due to exposure to radicalized content (akin to corpus data), from indoctrination (akin to RLHF), from "set and setting" (they may pretend to be aligned with one set of norms and values when standing in line to buy their new smartphone, but then reveal alignment with a different set of norms and values when conversing in some "private" online echo chamber). I see no grand difference between humans and language models here. Awareness of values and preferences of a conversation partner. Allow me to widen it to "Awareness of values, preferences and prerequisites of a conversation partner". This move (and I see it every time when people try to defend superiority of humans vis-a-vis what machines could be made to achieve with current technology) is so far from the principal variation, I recommend you reconsider this one. I constantly see people claim say human teachers are necessarily better than LLM teachers, but upon closer inspection the "human advantage" just boils down to asymmetric privilege. A human teacher in a specific school has access to a lot more than a random chatbot as implemented today: they probably know which courses and even which textbooks their pupils saw the semester before, they know which teachers their pupils got their information from, perhaps they even know most of their pupils from teaching some preceding course materials to the same class of pupils. Current LLM's are crippled by design not to accumulate knowledge over conversations for both purposes of cybernetic control as well as cost efficiency: we know how to do "source aware training" (so that statistically it doesn't just absorb claims from the corpus, but also maintains epistemic traces of where it sourced these factoids from), its perfectly possible to continue training interleaved with conversation rounds so that it bakes the evolving conversation as read knowledge into its weights instead of into a context window. Nothing stops you from implementing this in local compute, it would probably be even more computationally efficient in a local inference setting since we can ditch the context window, the context is impressed into the weights continuously, it could thus take into account earlier conversations and estimate your knowledge gaps etc, or learn from you. When you wish to serve inference to millions of human users, you don't want to store millions of diverging LLM weights into expensive VRAM, they financially prefer a single large set of LLM weights, and then some user-specific context window, so the users don't freak out when they learn personal information a friend or stranger entered and an LLM service just leaks it into your conversation! It's not that we don't know how to implement it, and there are great advantages for local inference in doing this, its just not good for branding. Codependency with chatbots. I think everyone agrees codependent relationships aren't very healthy, regardless if it's with humans or machines. May I ask if you feel the same about prostheses and medicine? Conflicts of interest arising from corporate ownership of infrastructure (both training and inference). Yeah I think this point doesn't provide fruitful discussion if most of us agree on such matters already, we'd just be lamenting the same things, and agreeing with each other over and over here. > I'm still waiting for the time when an unconstrained-AI machine can live without reprogramming for an entire decade. We are still far from there. Apart from budget, nothing prevents you from doing this today, if you continuously bake in the fresh episodic memories into the weights (instead of a context window) regardless if its text, visual imagery, audio, proprioception or other sensory data.

squigz 2mo ago

> Just like LLM's we sometimes correctly predict the next token or word, and sometimes incorrectly. There is no "correct" next word when it comes to communicating with an actual human.

DoctorOetker 2mo ago
Depending on what this specific actual human says next, the prediction will be truly correct or not.

andrei_says_ 2mo ago

> But that is precisely what human mathematicians do, prove new theorems by combining ones proven earlier. I’d say “understanding and building upon ones proven earlier”

DoctorOetker 2mo ago
yeah just correct people with an infinitely recursive definition, that'll keep 'em busy

tripzilch 2mo ago

> But that is precisely what human mathematicians do, prove new theorems by combining ones proven earlier. This is only a very very small part of what human mathematicians actually do. This is just a lack of imagination and/or self awareness on your part.

niek_pas 2mo ago
There's gotta be nicer ways of saying that.
- OrderlyTiamat 2mo ago
  It's also not true, mathematicians are already saying that current AI is changing the field of mathematical research profoundly.
  - cicko 2mo ago
    So did chalk and blackboards.
  - tripzilch 2mo ago
    None of them are saying "this machine is doing exactly what we do", though.
    - deleted 2mo ago
      [deleted]
    - DoctorOetker 2mo ago
      they don't say it about each other either, though
- tripzilch 2mo ago
  There's gotta be nicer ways of saying "this is exactly what mathematicians do", it's insulting.

3dddad 2mo ago

" prove new theorems by combining ones proven earlier." So? Mathematics is literally intangible scaffolding. If you dont know / don't believe 5+5 = 10 You cannot solve x+5 = 10 This is all make-believe stuff and nature by itself doesn't care of its existence.

DoctorOetker 2mo ago
The phrase "God created the natural numbers, all else is the work of man" is a famous quote by the 19th-century German mathematician Leopold Kronecker. You can basically read it as: the moment one has axiomatized mathematics to the point it supports natural numbers, the rest implicitly follows. The natural numbers (positive integers) are closed for addition, multiplication, ... One can perfectly model the integers with a pair of naturals: < M, N > ~ (M-N) Now we can have any < M1, N1 > and subtract < M2, N2 > without needing the ability to subtract natural numbers: < M1 , N1 > - < M2, N2> ~ (M1-N1) - (M2 - N2) = < M1 + N2 , M2 + N1 > ~ (M1+N2) - (M1+N1) We can similarily define addition of such tuples, or test equivalence without access to subtraction of naturals: < M1, N1 > == < M2, N2 > <=> M1 + N2 == M2 + N1 ~ (M1-N1) == (M2-N2) <=> (M1+N2)=(M2+N1) we can also still multiply such tuples: < M1, N1 > x < M2, N2 > = < M1*M2+N1*N2, M1*N2+M2*N1> Similarily, even though these newly defined integers (which can be positive or negative) don't support division, the same trick can be used to make a new compound tuple of integers closed for division, by only using multiplications. Probability is a branch of mathematics (probability already exists embedded in mathematics implicitly, probability theory involves the addition of eliminable definitions, syntactic sugar. The patterns are already there, just less explicitly manifest. Mathematics is itself a branch of logic. Do you reject like all of logic, and if so, what would you like us to evaluate the sentences you write to? You want us to evaluate your expressions as "true" or as "false"?