Dhruv Bansal

Physicist turned serial entrepreneur. Formerly in big data, currently building financial services for bitcoiners @unchainedcom. One day l'm going to space

[Dhruv Bansal @drhuvbansal]

Yesterday i expected a crash but we are holding the line #Bitcoin

[Parker Lewis @parkeralewis]

This event is perfect for any Silicon Valley investors in town for sxsw interested to learn about bitcoin and blockchain

[Dhruv Bansal @drhuvbansal]

Todays the day for some executive orders. Got a feeling actual XRP crypto talk will go down from the pres #Ripple

[Dhruv Bansal @drhuvbansal]

#Bitcoin miners don't do marketing or sales.

The Bitcoin network is a peer-to-peer market that allows miners to directly sell their product (hashrate) to their customers (us!) in exchange for bitcoin.

This is what "decentralized" means!

[Dhruv Bansal @drhuvbansal]

It bugs me when people don't understand #Bitcoin but are sure it's a scam b/c they read one skeptical article about it. I wish people would learn basic facts. SMH.

BTW I don't understand physics but I'm sure that quantum computing is a scam. Here's a link to an article I read...

[Unchained @unchained]

The Bitcoin Frontier Episode 59: Debating bitcoin: Invention or discovery? + its future in space with @dhruvbansal and @Excellion

- Bitcoin vs fire, internet, mathematics
- Bitcoin, sidechains, or Muskcoin on Mars
- Does digital scarcity truly exist?

[Dhruv Bansal @drhuvbansal]

Proof-of-stake? More like proof-of-mistake yuk yuk

[Dhruv Bansal @drhuvbansal]

Further evidence that miners *purchase* #Bitcoin and do not "create" it:

BCH miners "create" as just many coins per day as BTC miners but use ~200x less energy.

Is it because BCH mining is 200x more efficient than BTC?

No, it's because BCH tokens are 200x *less valuable*!

[Dhruv Bansal @drhuvbansal]

Altcoins today are valued on transaction volume and speed, accessibility, integrations, DAUs, developer count, &c. – metrics more commonly used to value tech startups than currencies.

[Dhruv Bansal @drhuvbansal]

Yes, it *is* possible for public companies to self-custody BTC while respecting compliance, audit, business continuity, and tax requirements all *without* creating a single point of failure or personal security risks for executives.

Yes, it's hard. Best practices often are

[Dhruv Bansal @drhuvbansal]

Imagine that #Bitcoin was released with a finite supply, proof-of-work, the difficulty readjustment, and the blockchain but NO Bitcoin Script, block size, or mempool. Coins exist (e.g. the coinbase), but transactions don't.

Would adding transactions require a soft or hard fork?

[Dhruv Bansal @drhuvbansal]

I recall reading somewhere about Satoshi defining an ever-increasing number of bitcoin transaction types to handle various use cases and then realizing that what was really required was a programmable virtual machine for transactions.

Where did I read this? Pls help, can't find

[Dhruv Bansal @drhuvbansal]

I bet it will turn out to be just some conversation I had once with Andrew Poelstra that I am misremembering as Satoshi's own words...someone prove me wrong.

[Dennis Porter @Dennis_Porter_]

We ❤️@unchainedcom.

Go see them for all your self-custody and multi-sig solutions.

[Parker Lewis @parkeralewis]

Just turned on a bricked bitcoin hardware wallet. Presumably the pressure from the faraday bag borked the screen. This is a lesson of why to keep backup recovery seeds in analog form, why to use multisig and why to periodically check your keys. A third party does not fix this!

[Brandon Quittem @Bquittem]

🚨 Minneapolis Bitcoiners Meetup CONFIRMED 🚨

WHEN: Tuesday, September 3rd @ 6PM

WHAT: Bitcoin-only meetup. Spread the word. Bring a friend.

@dhruvbansal is the keynote speaker, Co-Founder and CSO of @unchainedcom

WHERE: O'Shaughnessy Distillery in Minneapolis (upstairs)

DM me for an invite to the telegram group.

[QC @QiaochuYuan]

so i've been thinking a little bit about how i'd go about introducing someone to math, a few people in portland have expressed interest in something like this. the rest of this will be some messy thinking out loud. a major thing is that there's this large and imo very under-discussed gap between... let's say "high school math" and "advanced undergrad / intro grad school math" or something like that. like there's a big gap between, say, trigonometry and group theory. you need to traverse this gap if you want to understand the mathematics of quantum mechanics or relativity, for example, or a bunch of other things you might be interested in.

i've never heard anyone describe the nature of the gap in any real detail. a big part of what's difficult is that much of it is philosophical and ontological. in order to traverse the gap you have to go through multiple large conceptual revolutions in how you think about mathematics and what sort of a thing doing mathematics is or can be. you can tell this is actually difficult because historically it took a very long time and a lot of hard work for each of these revolutions to occur! they are not described explicitly and you somehow either pick them up by osmosis or don't. no language is introduced for any of this. so i think when students get stuck on one of these conceptual revolutions, it's very difficult for them to articulate what they're stuck on, and even if they could they might not have a teacher who can meaningfully address the issue.

i've been answering people's questions about math on the internet for about 15 years now, i've seen thousands and thousands of questions, and there's a particular set of sort of "classic confusing math stuff" that imo is related to the gap, and particularly to the *philosophical* nature of the gap. examples off the top of my head include:

1 = 0.999...
cantor's diagonal argument
what is a complex number
what are dx, dy, and dy/dx in calculus

there's a ton to say about all of these but i'll start with 1 = 0.999... because it's probably the most familiar one. so like, what's up with this? what the fuck does "..." really mean anyway? why isn't there an infinitely small gap between 1 and 0.999...?

these are good questions! i think people who get hung up on stuff like this worry that they're "bad at math" or something but imo it's the exact opposite! if you get hung up on stuff like this it means you are *actually thinking* about what these symbols are supposed to *mean* and you are getting confused about their *meaning*! that's great! there's an easy way to avoid getting hung up on stuff like this and it's to treat math like a symbolic game and not really think about meaning much either way. this is the opposite of doing mathematics!

when i get questions like this i try to congratulate people for even asking the question. it is delightful to me when people articulate philosophical confusions in mathematics, to my mind that's actually a very positive indicator that they'd do well and have a great time if only someone would properly explain to them what the hell is going on.

sadly, ime what people often get instead is like... "math gaslighting"? there's this kind of "well that's just the way it works and you just have to accept it" thing that some teachers apparently do which, again, to me is the opposite of doing mathematics. the pleasure of mathematics is getting to have the experience of seeing why a thing is true directly with your soul, seeing why it could not possibly be any other way, and in order to get that someone has to show you an argument that is convincing to you personally.

okay, so back to 1 = 0.999... there's a "standard party line" explaining how this is supposed to work in modern mathematics, and to go through it in full detail requires

1. an understanding of what a real number is according to modern mathematics (to get 100% clear on this requires understanding how the real numbers are constructed using set theory, and also why in some sense this construction does not matter)

2. an understanding of what it means to take the limit of a sequence of real numbers, so that you can discuss what it means to take an infinite sum of real numbers

3. a proof that the limit of the sequence 0.9, 0.99, 0.999, 0.9999, ... is exactly equal to 1 (this is in some sense the "easy part," the hard part is in steps 1 and 2 which set up the machinery to even allow us to express this).

there's a sort of sneaky underhanded ontological crime that gets committed along the way, involving the use of the term "real number." that term makes a very strong ontological claim - that the real numbers are the ones that really exist. this is actually extremely debatable - for example literally 100% of real numbers are non-computable in the sense that their decimal expansions can't be produced by any program. because there are only countably many programs, but uncountably many real numbers! even worse, the same argument appears to show that literally 100% of real numbers are non-describable - it's not possible to describe almost any of them in any way whatsoever. because there are only countably many descriptions! this argument turns out to run into significant set-theoretic subtleties but still, imo it's cause for concern. it's not clear that most of the real numbers actually exist in any meaningful sense! which is worrisome because all of physics and almost all of mathematics is nominally built on them!

the standard party line here about what the real numbers "really are" dates from about the end of the 19th century to the beginning of the 20th century and is rooted in the crisis in the foundation of mathematics, which is a whole other fucking thing. before that this was not how people thought about things (afaict). in the western tradition the concept of a real number is rooted firmly in euclidean geometry - it all starts with the length of a line segment - and euclidean geometry was not thought about in modern abstract terms but was (until general relativity!) thought to directly describe the geometric properties of physical space, as in the 3d space you and i inhabit and walk around in. you don't need any set theory to directly experience basic facts about physical space, such as: some things are long and rigid (like sticks), some of them are longer than others, you can put a long rigid thing next to another one to tell whether one is longer than the other, you can use a long rigid thing as a ruler to measure another long rigid thing and say "this one is 3 times as long as this other one," etc. etc.

math is supposed to have some sort of relationship to reality. in grade school we tell everyone that 2 + 2 = 4 is supposed to have something to do with how if johnny has 2 apples and susie has 2 apples and they put their apples together then they have 4 apples. and already once we get to 1 = 0.999... it's not clear what this is supposed to "really mean," according to any physical model of what numbers are. somehow a line segment is supposed to decompose into a piece that's 9/10ths as long as the original, another piece that's 9/100ths as long as the original, etc etc. but what does it really mean to cut up a line segment into infinitely many pieces? that's not something we can actually do! worrying about this kind of stuff goes all the way back to zeno and the formalism of infinite series does not really answer the philosophical question.

so, here's an example of one of the philosophical hurdles in the gap: in mathematics we work with "completed infinities." we manipulate infinite sets and reason about infinite processes all the time, and as part of mathematical training we learn a bunch of non-obvious stuff about how to do this without running into contradictions. there are good reasons to be philosophically skeptical about this, but also we've been doing it this way for a century or so now and not run into any problems in practice. quantum mechanics, quantum field theory, and relativity all require the real numbers as part of their foundations (QM and QFT even require the complex numbers) and are full of infinite sets and infinite processes and they work fine.

another philosophical hurdle is that you have to accept that the mathematical object called "the real numbers" is an adequate model of "the continuum," which properly refers to a pre-theoretical understanding of an aspect of physical space (the 1d aspect, having to do with lines and lengths and distances). even using this language "model" is breaking with standard mathematical practice - it's standard to just say that the continuum is the real numbers, full stop (e.g. this is what wikipedia says), which i think is a kind of ontological con. the continuum is pre-theoretical. you already know things about the continuum even if you have no mathematical training, just by virtue of existing in physical space.

so i think when people get hung up on 1 = 0.999..., part of what they might be hung up on involves these philosophical issues, which are not really recognized or respected, and which it's difficult to even find language for. i think it's completely reasonable to be hung up on whether it makes sense to talk about infinite processes, or whether the real numbers as defined in modern mathematics adequately model the continuum. for example, the real numbers don't contain any infinitely small or large elements, and that was a historically contingent decision! there are alternatives like the nonstandard reals that do, that can e.g. be used to rigorously do calculus using infinitesimals. and in some of those alternatives the sequence 0.9, 0.99, 0.999, ... no longer has a limit at all! the meaning of "0.999..." actually has to be reinterpreted to involve decimal expansions which are indexed by nonstandard integers (loosely speaking, these are also allowed to be infinitely large) in order to get something equal to 1 in this setting.

i see this all as part of a larger trend i think about periodically where at some point in the 20th century it seems like mathematics lost something, that has to do with this disconnect from philosophy and from the world. you can really feel it in the differences between how mathematicians wrote in the 19th century vs. now. i think about both of these quotes and take them increasingly seriously:

> In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics. — Hermann Weyl, 1939

> Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine. — Sir Michael Atiyah, 2002

[Spencer Greenberg 🔍 @SpencrGreenberg]

Does astrology work? We tested the ability of 152 astrologers to see if they could demonstrate genuine astrological skill.

Here is how the study was designed and what we found (including a result that really surprised me):

🧵

[Lisa Hough @lisa_hough_]

Don’t Sleep On This ⬇️⬇️⬇️⬇️⬇️⬇��

In 1999, Peter Thiel took a tiny bet—a $2,000 contribution to a Roth IRA—and turned it into a game-changer. He bought shares of his fledgling startup, PayPal, for a mere $0.001 each.

Fast-forward to 2002, when eBay buys PayPal for $1.5 billion. Thiel’s little IRA is no longer small change. But he didn’t stop there. He reinvested in startups, including a small company called Facebook.

Here’s the kicker: Roth IRAs mean tax-free growth. That tiny bet grew, multiplied, and exploded into a $5 billion treasure chest by 2021—all without a penny lost to taxes on the gains.

Be Like Peter. Take a small bet. Plant a seed. Watch it grow.

Your future self will thank you.