An approximate introduction to how zk-SNARKs are possible (part 2)
If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.
Polynomials
Polynomials are a special class of algebraic expressions of the form:
- x+5
- x^4
- x^3+3x^2+3x+1
- 628x^{271}+318x^{270}+530x^{269}+…+69x+381
i.e. they are a sum of any (finite!) number of terms of the form cx^k
There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.
Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:
- A(0)+B(0)=C(0)
- A(1)+B(1)=C(1)
- A(2)+B(2)=C(2)
- A(3)+B(3)=C(3)
And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:
- 12+1=13
- 10+8=18
- 15+8=23
- 15+13=28
You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:
- A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
- B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
- C(x)=5x+13
Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.
Comparing a polynomial to itself
You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)
As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.
Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).
Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!
Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.
see part 3
(Edited)

Hackernoon
4 years ago
👏 Awesome post! When is part 3 coming?

Trent Lapinski
4 years ago
Very complex topic, great explanation
More on Web3 & Crypto

Stephen Moore
3 years ago
Web 2 + Web 3 = Web 5.
Monkey jpegs and shitcoins have tarnished Web3's reputation. Let’s move on.
Web3 was called "the internet's future."
Well, 'crypto bros' shouted about it loudly.
As quickly as it arrived to be the next internet, it appears to be dead. It's had scandals, turbulence, and crashes galore:
Web 3.0's cryptocurrencies have crashed. Bitcoin's all-time high was $66,935. This month, Ethereum fell from $2130 to $1117. Six months ago, the cryptocurrency market peaked at $3 trillion. Worst is likely ahead.
Gas fees make even the simplest Web3 blockchain transactions unsustainable.
Terra, Luna, and other dollar pegs collapsed, hurting crypto markets. Celsius, a crypto lender backed by VCs and Canada's second-largest pension fund, and Binance, a crypto marketplace, have withheld money and coins. They're near collapse.
NFT sales are falling rapidly and losing public interest.
Web3 has few real-world uses, like most crypto/blockchain technologies. Web3's image has been tarnished by monkey profile pictures and shitcoins while failing to become decentralized (the whole concept is controlled by VCs).
The damage seems irreparable, leaving Web3 in the gutter.
Step forward our new saviour — Web5
Fear not though, as hero awaits to drag us out of the Web3 hellscape. Jack Dorsey revealed his plan to save the internet quickly.
Dorsey has long criticized Web3, believing that VC capital and silicon valley insiders have created a centralized platform. In a tweet that upset believers and VCs (he was promptly blocked by Marc Andreessen), Dorsey argued, "You don't own "Web3." VCs and LPs do. Their incentives prevent it. It's a centralized organization with a new name.
Dorsey announced Web5 on June 10 in a very Elon-like manner. Block's TBD unit will work on the project (formerly Square).
Web5's pitch is that users will control their own data and identity. Bitcoin-based. Sound familiar? The presentation pack's official definition emphasizes decentralization. Web5 is a decentralized web platform that enables developers to write decentralized web apps using decentralized identifiers, verifiable credentials, and decentralized web nodes, returning ownership and control over identity and data to individuals.
Web5 would be permission-less, open, and token-less. What that means for Earth is anyone's guess. Identity. Ownership. Blockchains. Bitcoin. Different.
Web4 appears to have been skipped, forever destined to wish it could have shown the world what it could have been. (It was probably crap.) As this iteration combines Web2 and Web3, simple math and common sense add up to 5. Or something.
Dorsey and his team have had this idea simmering for a while. Daniel Buchner, a member of Block's Decentralized Identity team, said, "We're finishing up Web5's technical components."
Web5 could be the project that decentralizes the internet. It must be useful to users and convince everyone to drop the countless Web3 projects, products, services, coins, blockchains, and websites being developed as I write this.
Web5 may be too late for Dorsey and the incoming flood of creators.
Web6 is planned!
The next months and years will be hectic and less stable than the transition from Web 1.0 to Web 2.0.
Web1 was around 1991-2004.
Web2 ran from 2004 to 2021. (though the Web3 term was first used in 2014, it only really gained traction years later.)
Web3 lasted a year.
Web4 is dead.
Silicon Valley billionaires are turning it into a startup-style race, each disrupting the next iteration until they crack it. Or destroy it completely.
Web5 won't last either.

Yogesh Rawal
3 years ago
Blockchain to solve growing privacy challenges
Most online activity is now public. Businesses collect, store, and use our personal data to improve sales and services.
In 2014, Uber executives and employees were accused of spying on customers using tools like maps. Another incident raised concerns about the use of ‘FaceApp'. The app was created by a small Russian company, and the photos can be used in unexpected ways. The Cambridge Analytica scandal exposed serious privacy issues. The whole incident raised questions about how governments and businesses should handle data. Modern technologies and practices also make it easier to link data to people.
As a result, governments and regulators have taken steps to protect user data. The General Data Protection Regulation (GDPR) was introduced by the EU to address data privacy issues. The law governs how businesses collect and process user data. The Data Protection Bill in India and the General Data Protection Law in Brazil are similar.
Despite the impact these regulations have made on data practices, a lot of distance is yet to cover.
Blockchain's solution
Blockchain may be able to address growing data privacy concerns. The technology protects our personal data by providing security and anonymity. The blockchain uses random strings of numbers called public and private keys to maintain privacy. These keys allow a person to be identified without revealing their identity. Blockchain may be able to ensure data privacy and security in this way. Let's dig deeper.
Financial transactions
Online payments require third-party services like PayPal or Google Pay. Using blockchain can eliminate the need to trust third parties. Users can send payments between peers using their public and private keys without providing personal information to a third-party application. Blockchain will also secure financial data.
Healthcare data
Blockchain technology can give patients more control over their data. There are benefits to doing so. Once the data is recorded on the ledger, patients can keep it secure and only allow authorized access. They can also only give the healthcare provider part of the information needed.
The major challenge
We tried to figure out how blockchain could help solve the growing data privacy issues. However, using blockchain to address privacy concerns has significant drawbacks. Blockchain is not designed for data privacy. A ‘distributed' ledger will be used to store the data. Another issue is the immutability of blockchain. Data entered into the ledger cannot be changed or deleted. It will be impossible to remove personal data from the ledger even if desired.
MIT's Enigma Project aims to solve this. Enigma's ‘Secret Network' allows nodes to process data without seeing it. Decentralized applications can use Secret Network to use encrypted data without revealing it.
Another startup, Oasis Labs, uses blockchain to address data privacy issues. They are working on a system that will allow businesses to protect their customers' data.
Conclusion
Blockchain technology is already being used. Several governments use blockchain to eliminate centralized servers and improve data security. In this information age, it is vital to safeguard our data. How blockchain can help us in this matter is still unknown as the world explores the technology.

The Verge
3 years ago
Bored Ape Yacht Club creator raises $450 million at a $4 billion valuation.
Yuga Labs, owner of three of the biggest NFT brands on the market, announced today a $450 million funding round. The money will be used to create a media empire based on NFTs, starting with games and a metaverse project.
The team's Otherside metaverse project is an MMORPG meant to connect the larger NFT universe. They want to create “an interoperable world” that is “gamified” and “completely decentralized,” says Wylie Aronow, aka Gordon Goner, co-founder of Bored Ape Yacht Club. “We think the real Ready Player One experience will be player run.”
Just a few weeks ago, Yuga Labs announced the acquisition of CryptoPunks and Meebits from Larva Labs. The deal brought together three of the most valuable NFT collections, giving Yuga Labs more IP to work with when developing games and metaverses. Last week, ApeCoin was launched as a cryptocurrency that will be governed independently and used in Yuga Labs properties.
Otherside will be developed by “a few different game studios,” says Yuga Labs CEO Nicole Muniz. The company plans to create development tools that allow NFTs from other projects to work inside their world. “We're welcoming everyone into a walled garden.”
However, Yuga Labs believes that other companies are approaching metaverse projects incorrectly, allowing the startup to stand out. People won't bond spending time in a virtual space with nothing going on, says Yuga Labs co-founder Greg Solano, aka Gargamel. Instead, he says, people bond when forced to work together.
In order to avoid getting smacked, Solano advises making friends. “We don't think a Zoom chat and walking around saying ‘hi' creates a deep social experience.” Yuga Labs refused to provide a release date for Otherside. Later this year, a play-to-win game is planned.
The funding round was led by Andreessen Horowitz, a major investor in the Web3 space. It previously backed OpenSea and Coinbase. Animoca Brands, Coinbase, and MoonPay are among those who have invested. Andreessen Horowitz general partner Chris Lyons will join Yuga Labs' board. The Financial Times broke the story last month.
"META IS A DOMINANT DIGITAL EXPERIENCE PROVIDER IN A DYSTOPIAN FUTURE."
This emerging [Web3] ecosystem is important to me, as it is to companies like Meta,” Chris Dixon, head of Andreessen Horowitz's crypto arm, tells The Verge. “In a dystopian future, Meta is the dominant digital experience provider, and it controls all the money and power.” (Andreessen Horowitz co-founder Marc Andreessen sits on Meta's board and invested early in Facebook.)
Yuga Labs has been profitable so far. According to a leaked pitch deck, the company made $137 million last year, primarily from its NFT brands, with a 95% profit margin. (Yuga Labs declined to comment on deck figures.)
But the company has built little so far. According to OpenSea data, it has only released one game for a limited time. That means Yuga Labs gets hundreds of millions of dollars to build a gaming company from scratch, based on a hugely lucrative art project.
Investors fund Yuga Labs based on its success. That's what they did, says Dixon, “they created a culture phenomenon”. But ultimately, the company is betting on the same thing that so many others are: that a metaverse project will be the next big thing. Now they must construct it.
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Dr Mehmet Yildiz
3 years ago
How I train my brain daily for clarity and productivity.
I use a conceptual and practical system I developed decades ago as an example.
Since childhood, I've been interested in the brain-mind connection, so I developed a system using scientific breakthroughs, experiments, and the experiences of successful people in my circles.
This story provides a high-level overview of a custom system to inform and inspire readers. Creating a mind gym was one of my best personal and professional investments.
Such a complex system may not be possible for everyone or appear luxurious at first. However, the process and approach may help you find more accessible and viable solutions.
Visualizing the brain as a muscle, I learned to stimulate it with physical and mental exercises, applying a new mindset and behavioral changes.
My methods and practices may not work for others because we're all different. I focus on the approach's principles and highlights so you can create your own program.
Some create a conceptual and practical system intuitively, and others intellectually. Both worked. I see intellect and intuition as higher selves.
The mental tools I introduce are based on lifestyle changes and can be personalized by anyone, barring physical constraints or underlying health conditions.
Some people can't meditate despite wanting to due to mental constraints. This story lacks exceptions.
People's systems may vary. Many have used my tools successfully. All have scientific backing because their benefits attracted scientists. None are unethical or controversial.
My focus is cognition, which is the neocortex's ability. These practices and tools can affect the limbic and reptilian brain regions.
A previous article discussed brain health's biological aspects. This article focuses on psychology.
Thinking, learning, and remembering are cognitive abilities. Cognitive abilities determine our health and performance.
Cognitive health is the ability to think, concentrate, learn, and remember. Cognitive performance boosting involves various tools and processes. My system and protocols address cognitive health and performance.
As a biological organ, the brain's abilities decline with age, especially if not used regularly. Older people have more neurodegenerative disorders like dementia.
As aging is inevitable, I focus on creating cognitive reserves to remain mentally functional as we age and face mental decline or cognitive impairment.
My protocols focus on neurogenesis, or brain growth and maintenance. Neurons and connections can grow at any age.
Metacognition refers to knowing our cognitive abilities, like thinking about thinking and learning how to learn.
In the following sections, I provide an overview of my system, mental tools, and protocols.
This system summarizes my 50-year career. Some may find it too abstract, so I give examples.
First, explain the system. Section 2 introduces activities. Third, how to measure and maintain mental growth.
1 — Developed a practical mental gym.
The mental gym is a metaphor for the physical fitness gym to improve our mental muscles.
This concept covers brain and mind functionality. Integrated biological and psychological components.
I'll describe my mental gym so my other points make sense. My mental gym has physical and mental tools.
Mindfulness, meditation, visualization, self-conversations, breathing exercises, expressive writing, working in a flow state, reading, music, dance, isometric training, barefoot walking, cold/heat exposure, CBT, and social engagements are regular tools.
Dancing, walking, and thermogenesis are body-related tools. As the brain is part of the body and houses the mind, these tools can affect mental abilities such as attention, focus, memory, task switching, and problem-solving.
Different people may like different tools. I chose these tools based on my needs, goals, and lifestyle. They're just examples. You can choose tools that fit your goals and personality.
2 — Performed tasks regularly.
These tools gave me clarity. They became daily hobbies. Some I did alone, others with others.
Some examples: I meditate daily. Even though my overactive mind made daily meditation difficult at first, I now enjoy it. Meditation three times a day sharpens my mind.
Self-talk is used for self-therapy and creativity. Self-talk was initially difficult, but neurogenesis rewired my brain to make it a habit.
Cold showers, warm baths with Epsom salts, fasting, barefoot walks on the beach or grass, dancing, calisthenics, trampoline hopping, and breathing exercises increase my mental clarity, creativity, and productivity.
These exercises can increase BDNF, which promotes nervous system growth. They improve mental capacity and performance by increasing blood flow and brain oxygenation.
I use weekly and occasional activities like dry saunas, talking with others, and community activities.
These activities stimulate the brain and mind, improving performance and cognitive capacity.
3 — Measured progress, set growth goals.
Measuring progress helps us stay on track. Without data, it's hard to stay motivated. When we face inevitable setbacks, we may abandon our dreams.
I created a daily checklist for a spreadsheet with macros. I tracked how often and long I did each activity.
I measured my progress objectively and subjectively. In the progress spreadsheet, I noted my meditation hours and subjective feelings.
In another column, I used good, moderate, and excellent to get qualitative data. It took time and effort. Later, I started benefiting from this automated structure.
Creating a page for each activity, such as meditation, self-talk, cold showers, walking, expressive writing, personal interactions, etc., gave me empirical data I could analyze, modify, and graph to show progress.
Colored charts showed each area's strengths and weaknesses.
Strengths motivate me to continue them. Identifying weaknesses helped me improve them.
As the system matured, data recording became a habit and took less time. I saw the result immediately because I automated the charts when I entered daily data. Early time investment paid off later.
Mind Gym Benefits, Effective Use, and Progress Measuring
This concept helped me move from comfort to risk. I accept things as they are.
Turnarounds were made. I stopped feeling "Fight-Flight-Freeze" and maintained self-control.
I tamed my overactive amygdala by strengthening my brain. Stress and anxiety decreased. With these shifts, I accepted criticism and turned envy into admiration. Clarity improved.
When the cognitive part of the brain became stronger and the primitive part was tamed, managing thoughts and emotions became easier. My AQ increased. I learned to tolerate people, physical, mental, and emotional obstacles.
Accessing vast information sources in my subconscious mind through an improved RAS allowed me to easily tap into my higher self and recognize flaws in my lower self.
Summary
The brain loves patterns and routines, so habits help. Observing, developing, and monitoring habits mindfully can be beneficial. Mindfulness helps us achieve this goal systematically.
As body and mind are connected, we must consider both when building habits. Consistent and joyful practices can strengthen neurons and neural connections.
Habits help us accomplish more with less effort. Regularly using mental tools and processes can improve our cognitive health and performance as we age.
Creating daily habits to improve cognitive abilities can sharpen our minds and boost our well-being.
Some apps monitor our activities and behavior to help build habits. If you can't replicate my system, try these apps. Some smartwatches and fitness devices include them.
Set aside time each day for mental activities you enjoy. Regular scheduling and practice can strengthen brain regions and form habits. Once you form habits, tasks become easy.
Improving our minds is a lifelong journey. It's easier and more sustainable to increase our efforts daily, weekly, monthly, or annually.
Despite life's ups and downs, many want to remain calm and cheerful.
This valuable skill is unrelated to wealth or fame. It's about our mindset, fueled by our biological and psychological needs.
Here are some lessons I've learned about staying calm and composed despite challenges and setbacks.
1 — Tranquillity starts with observing thoughts and feelings.
2 — Clear the mental clutter and emotional entanglements with conscious breathing and gentle movements.
3 — Accept situations and events as they are with no resistance.
4 — Self-love can lead to loving others and increasing compassion.
5 — Count your blessings and cultivate gratitude.
Clear thinking can bring joy and satisfaction. It's a privilege to wake up with a healthy body and clear mind, ready to connect with others and serve them.
Thank you for reading my perspectives. I wish you a healthy and happy life.

Ossiana Tepfenhart
3 years ago
Has anyone noticed what an absolute shitshow LinkedIn is?
After viewing its insanity, I had to leave this platform.
I joined LinkedIn recently. That's how I aim to increase my readership and gain recognition. LinkedIn's premise appealed to me: a Facebook-like platform for professional networking.
I don't use Facebook since it's full of propaganda. It seems like a professional, apolitical space, right?
I expected people to:
be more formal and respectful than on Facebook.
Talk about the inclusiveness of the workplace. Studies consistently demonstrate that inclusive, progressive workplaces outperform those that adhere to established practices.
Talk about business in their industry. Yep. I wanted to read articles with advice on how to write better and reach a wider audience.
Oh, sh*t. I hadn't anticipated that.
After posting and reading about inclusivity and pro-choice, I was startled by how many professionals acted unprofessionally. I've seen:
Men have approached me in the DMs in a really aggressive manner. Yikes. huge yikes Not at all professional.
I've heard pro-choice women referred to as infant killers by many people. If I were the CEO of a company and I witnessed one of my employees acting that poorly, I would immediately fire them.
Many posts are anti-LGBTQIA+, as I've noticed. a lot, like, a lot. Some are subtly stating that the world doesn't need to know, while others are openly making fun of transgender persons like myself.
Several medical professionals were posting explicitly racist comments. Even if you are as white as a sheet like me, you should be alarmed by this. Who's to guarantee a patient who is black won't unintentionally die?
I won't even get into how many men in STEM I observed pushing for the exclusion of women from their fields. I shouldn't be surprised considering the majority of those men I've encountered have a passionate dislike for women, but goddamn, dude.
Many people appear entirely too at ease displaying their bigotry on their professional profiles.
As a white female, I'm always shocked by people's open hostility. Professional environments are very important.
I don't know if this is still true (people seem too politicized to care), but if I heard many of these statements in person, I'd suppose they feel ashamed. Really.
Are you not ashamed of being so mean? Are you so weak that competing with others terrifies you? Isn't this embarrassing?
LinkedIn isn't great at censoring offensive comments. These people aren't getting warnings. So they were safe while others were unsafe.
The CEO in me would want to know if I had placed a bigot on my staff.
I always wondered if people's employers knew about their online behavior. If they know how horrible they appear, they don't care.
As a manager, I was picky about hiring. Obviously. In most industries, it costs $1,000 or more to hire a full-time employee, so be sure it pays off.
Companies that embrace diversity and tolerance (and are intolerant of intolerance) are more profitable, likely to recruit top personnel, and successful.
People avoid businesses that alienate them. That's why I don't eat at Chic-Fil-A and why folks avoid MyPillow. Being inclusive is good business.
CEOs are harmed by online bigots. Image is an issue. If you're a business owner, you can fire staff who don't help you.
On the one hand, I'm delighted it makes it simpler to identify those with whom not to do business.
Don’t get me wrong. I'm glad I know who to avoid when hiring, getting references, or searching for a job. When people are bad, it saves me time.
What's up with professionalism?
Really. I need to know. I've crossed the boundary between acceptable and unacceptable behavior, but never on a professional platform. I got in trouble for not wearing bras even though it's not part of my gender expression.
If I behaved like that at my last two office jobs, my supervisors would have fired me immediately. Some of the behavior I've seen is so outrageous, I can't believe these people have employment. Some are even leaders.
Like…how? Is hatred now normalized?
Please pay attention whether you're seeking for a job or even simply a side gig.
Do not add to the tragedy that LinkedIn comments can be, or at least don't make uninformed comments. Even if you weren't banned, the site may still bite you.
Recruiters can and do look at your activity. Your writing goes on your résumé. The wrong comment might lose you a job.
Recruiters and CEOs might reject candidates whose principles contradict with their corporate culture. Bigotry will get you banned from many companies, especially if others report you.
If you want a high-paying job, avoid being a LinkedIn asshole. People care even if you think no one does. Before speaking, ponder. Is this how you want to be perceived?
Better advice:
If your politics might turn off an employer, stop posting about them online and ask yourself why you hold such objectionable ideas.

James White
3 years ago
Three Books That Can Change Your Life in a Day
I've summarized each.
Anne Lamott said books are important. Books help us understand ourselves and our behavior. They teach us about community, friendship, and death.
I read. One of my few life-changing habits. 100+ books a year improve my life. I'll list life-changing books you can read in a day. I hope you like them too.
Let's get started!
1) Seneca's Letters from a Stoic
One of my favorite philosophy books. Ryan Holiday, Naval Ravikant, and other prolific readers recommend it.
Seneca wrote 124 letters at the end of his life after working for Nero. Death, friendship, and virtue are discussed.
It's worth rereading. When I'm in trouble, I consult Seneca.
It's brief. The book could be read in one day. However, use it for guidance during difficult times.
My favorite book quotes:
Many men find that becoming wealthy only alters their problems rather than solving them.
You will never be poor if you live in harmony with nature; you will never be wealthy if you live according to what other people think.
We suffer more frequently in our imagination than in reality; there are more things that are likely to frighten us than to crush us.
2) Steven Pressfield's book The War of Art
I’ve read this book twice. I'll likely reread it before 2022 is over.
The War Of Art is the best productivity book. Steven offers procrastination-fighting tips.
Writers, musicians, and creative types will love The War of Art. Workplace procrastinators should also read this book.
My favorite book quotes:
The act of creation is what matters most in art. Other than sitting down and making an effort every day, nothing else matters.
Working creatively is not a selfish endeavor or an attempt by the actor to gain attention. It serves as a gift for all living things in the world. Don't steal your contribution from us. Give us everything you have.
Fear is healthy. Fear is a signal, just like self-doubt. Fear instructs us on what to do. The more terrified we are of a task or calling, the more certain we can be that we must complete it.
3) Darren Hardy's The Compound Effect
The Compound Effect offers practical tips to boost productivity by 10x.
The author believes each choice shapes your future. Pizza may seem harmless. However, daily use increases heart disease risk.
Positive outcomes too. Daily gym visits improve fitness. Reading an hour each night can help you learn. Writing 1,000 words per day would allow you to write a novel in under a year.
Your daily choices affect compound interest and your future. Thus, better habits can improve your life.
My favorite book quotes:
Until you alter a daily habit, you cannot change your life. The key to your success can be found in the actions you take each day.
The hundreds, thousands, or millions of little things are what distinguish the ordinary from the extraordinary; it is not the big things that add up in the end.
Don't worry about willpower. Time to use why-power. Only when you relate your decisions to your aspirations and dreams will they have any real meaning. The decisions that are in line with what you define as your purpose, your core self, and your highest values are the wisest and most inspiring ones. To avoid giving up too easily, you must want something and understand why you want it.
