There is a lot of arguing about the history of numerology. Different groups like to carry on about how they have the "real true numerology," and they clutch this idea very tightly to their horrid little hearts.
Pythagoras is often regarded as the father of numerology. It is true that he regarded numbers as both mystical and practical forces. He was interested in numbers and their vibrations with respect to things like colours and musical notes, though. He wasn't sitting down, scroll in hand, bothering himself with life paths and name analysis. He simply wasn't doing those things.
That didn't happen until many centuries later, when an occultist named Heinrich Cornelius Agrippa took the Aramaic alphanumeric key, which had long since been applied to the Hebrew alphabet and used by Hebrew mystics and scholars for gematria (the numerology of words). Agrippa mapped the numbers 1 to 9 onto the Latin alphabet as it was in the 1500s. "Pythagorean Numerology" would be better called "Agrippan Numerology," but 99.9% of the New Age community has never heard of him. That alphabet key is not exactly what we are familiar with today, but it is close.
In today's modern world, the numerology we call "Pythagorean" comes from a woman named Sarah Balliet from New Jersey, who wrote books under the name Mrs. L. Dow Balliet. Her good friend and fellow number mystic, Dr. Julia Seton, MD, coined the term "numerology", probably because saying "The Balliet System of the Tone and Vibration of Numbers" was a bit cumbersome.
Sarah Balliet was no doubt influenced by Agrippa and Pythagoras.
At the same time, across the ocean in Europe, a much more flamboyant personality was spreading the good news about mystical math. Cheiro, a pseudonym for William John Warner, an Irish astrologer, numerologist, and palmist who asserted that he learned all that he knew from Brahmin teachers in India. He called his numerology "Chaldean". The calculations and interpretations are a little different from the Western Pythagorean ones. The alphabet key is also different; the letters are placed with numbers based on their sound, not on their place in the alphabet. The number nine is also not used in the Chaldean alphabet key.
At the same time, the astrologer Sepharial (Walter Gorm Old) was writing about numerology, using formulas similar to Cheiro. His alphabet key was somewhat different, as it assigned phonetic sounds like "tz" and "th" numerical values and included the number nine.
The Church of Light, based in the USA, founded by astrologer and occultist C.C. Zain (Benjamin Parker Williams) uses an alphabet key similar to Sepharial's and has even printed it on their Tarot cards, the Brotherhood of Light Tarot Deck. "Sacred Tarot", still being published by the organization, is a companion book to these cards.
There are many more authors from the 1880s to the 1930s who wrote about and influenced numerology. Over the years, their work has faded, and we have gone from "this work was influenced by Pythagoras and Agrippa" to "This is the undisputed and unbroken super secret teachings of Pythagoras himself" on too many websites hawking too many courses to count.
Numerology is modern. What we use today is the result of several occultists and mystics in Europe, the UK, and the USA (and other places, too, no doubt!) experimenting, watching patterns play out in the real world, and going back to the drawing board when their predictions did not pan out. It is not an unbroken line back to Pythagoras, the ancient Egyptians, or the Atlanteans. It is modern, and much of what the public has been exposed to through mass-market publications is derived from the work of Sarah Balliet, Dr. Julia Seton, MD, and her daughter, Dr. Juno Jordan, PhD.
The same can be said for the Chaldean and Vedic branches of numerology. Either Vedic numerology is three Chaldean numerologies in a trenchcoat or Chaldean numerology is three Vedic numerologies in a trenchcoat -- they are almost indistinguishable from each other. While ancient Indian texts like the Sankhya Shastra deal with the philosophy of numbers, there is no evidence in the Vedas of a system for calculating your Life Path number, Day Number, Personal Year, or any other placement from either your birthday or your name, to predict your career, personality, or anything else.
So here is how I approach it, and it always seems to irritate a lot of the other numerologists: It's all valid. There is no such thing as the "wrong" numerology. If Western systems resonate with you, study them. If you prefer Eastern systems, jump in. I teach both, and prefer using the Vedic ank. They both take different roads, but they arrive at the same place.
Being modern doesn't make it wrong. It's actually exciting because 150 years is still really YOUNG for a system of metaphysics. It puts us at the helm, creating the body of work that people will use for centuries to come.