善其事、利其器；以數為例

所謂工欲善其事必先利其器，正如庖丁解牛，又像如魚得水。要做好一件事，要有合適的人選及合適的工具，好讓能人可善事，利器得善用，相得益彰也。下圖是用AI Gemini創作。

我謹以數字為例，提問如果沒有阿拉伯數字，來引申出數學發展必受限制，望能點出其中的精緒，並刺激大家思考當中的問題和道理。有不少人相信如果沒有阿拉伯數字（0 到 9的數值記數法），今天人類的日常生活、科學計算與金融體系將會截然不同。何解？

讓我們嘗試以漢字記數法作個說明，假設今天中國只有漢字數字，即中文的一、二、三、‧‧‧、九，或古代中國的算籌與帳碼。這樣一來，我們如何寫下算式如： 123+654呢？又如何計算51x119呢？如此類推，大家大概開始明白阿拉伯數字的簡易及便利性實在是難以取代的！

如此類推，如歐洲如今天仍只有羅馬數字，即 I, V, X, L, C, D, M 等來記數，但缺乏「零」，他們的算術運算如加減乘除會如何處理呢？其他文化體系的記數法，如古埃及數字對記數以至數學發展又會有何影響呢？

如果沒有阿拉伯數字，對社會與科技的影響，大概會大大減慢數學的發展，例如大數據、分數、無理數(irrational numbers)以至微積分運算會變得非常笨重，甚或無法被發展出來，而工程建構和商業記帳等運作便需要花費倍增的時間與人力。當然，這只是憑簡單及直覺的推論，若要做足科學論證來比試阿拉伯數字之有無對世上各國在數學上的影響，實非小弟及本文所能涵蓋的課題，本文旨在以阿拉伯數字的簡易及方便性來帶出工欲善其事，必先利其器之一例。

事實上，要做好諸如數學的研究，外需要有利之器，內則需要有人之理性協調能力，二者缺一不可！或說，人類認知思維可說是廣闊無邊的，要用好這思維之效用，總要在無邊的意念中作出過濾和沉澱，好形成一個實踐之臺階，相關利器便是這個臺階，尤如腳所能踏之實地。只有人而無實地，無以建家立業，只有實地而無人，則無經營創建之可能。

朋友們，在你的專長或專業當中，有類似之善其事而先利其器的情況嗎？你有信心善用你的人生嗎？

The Mystical experience of Thomas Aquinas

Thomas Acquinas is not only a Doctor of the Church, but the “Angelic Doctor,” as the Dominicans like to say. His works are vast and overwhelming. Indeed, he is a highly respected and influential theologian in the Catholic Church. The illustration figure is generated by AI Domini.

On the feast of St. Nicholas [in 1273], St. Thomas was celebrating Mass when he received a spiritual revelation that so affected him. Consequently, he stopped his writing work leaving his great work the ‘Summa Theologiae’ unfinished. To Brother Reginald’s (his secretary and friend) expostulations, he replied, ‘The end of my labors has come. All that I have written seems to me like straw compared to what I have seen and what has been revealed to me.’ Aquinas later said, ‘I can write no more. I have seen things that make my writings like straw.’

In his mystical event, Aquinas entered a deep and unusually long state of ecstasy or divine contemplation while at the altar, The "straw" remark could be understood to mean that the Divine revelation in question is truly transcendental and ineffable. His experience exposed the limits of human language. Aquinas humbly acknowledged that human concepts and words are utterly inadequate to capture the direct, transcendent glory of God.

Aquinas famously held that "grace does not destroy nature, but perfects it" (gratia non tollit naturam, sed perficit). In his view, human reason is a gift from God capable of discovering true things about creation and even proving certain truths about God (like His existence). Reason, however, has strict limits. For Aquinas, human intellect is a stepping stone. It climbs as high as it possibly can through logical arguments, but it eventually reaches a ceiling where it must hand the reins over to faith and divine revelation. In short, Aquinas used reason to build a magnificent cathedral of thought. However, once he encountered the Almighty in spirit, the Divine revelation was too profound and transcendental to be expressed in words available in his reach!

To sum up, Aquinas argued that faith and reason come from the same divine source and cannot contradict each other. Reason can establish certain truths about God (natural theology), while supernatural mysteries surpass human reason and require divine revelation. For believers in Christ, it becomes a matter of faith and choice. Would you agree with Aquinas that God’s truth is really beyond words? Have you come across a similar issue in your life?

In fact, Aquinas is just one of the Church Fathers who had a similar mystical experience of seeing or meeting God. Additionally, the recent study of near-death heaven experiences strengthens our belief that the fullness of God is truly supernatural and transcendental beyond our intellect and description. As such, our faith in God should accordingly be extended to embrace such ineffable mystery. Nevertheless, it should not exempt us from using our gifts (e.g. wisdom and free will) wholeheartedly and respectfully in our quest to understand and be united with the Lord in our earthly journeys. May God help us, Amen!

©Howard the Evangelist All rights reserved

Reproduction of this material, in whole or in part, is permitted for non-commercial, educational, or personal use, provided that the source is clearly acknowledged and cited.

An infinite multitude of universes as perceived by an NDEr

In 2017, Philip Hasheider, an accomplished author and farmer, unexpectedly collapsed during a workout, resulting in a near-death experience that transported him to an infinite, timeless dimension. He encountered a radiant golden sphere—symbolizing the Divine Creator—radiating love and connected to an infinite web of universes. After being revived, Philip spent months grappling with the meaning of his experience, eventually realizing his purpose: to be a conduit of hope for those who feel lost.

In another interview titled Clinically Dead 6 Minutes; Man Visits Infinite Universe of 8 Dec, 2024, he had the following ideas and understanding of the infinite and timeless dimension in question: Imagining that we could take our universe and press it down to a single sheet of paper (say) and suspend it out there in that eternal & timeless space. Let’s suppose we could also make similar treatments for the innumerable universes therein. Then, each one of these universes (sheets) would be seen to have a filament reaching up and attaching itself to the God Essence energy which is the pulsating golden sphere seen by him in his NDE. The God Essence energy was sending all the love, all the welcoming and all the acceptance (or any similar adjective) into all these different universes! Such a conglomerate of the universes form a spider web-like matrix which encompasses the golden sphere.

Indeed, Philip had some hesitation in his description of the infinite and timeless dimension since its true qualities are really ineffable. He could only try his best to convey his memories into words and ideas for our understanding as limited by our physical perception. Such a phenomenon is rather common among near-death experiencers when they are asked about what exactly they had seen and experienced.

Similarly, let’s try to expand our 3D to a 4D world while time is unchanged. While we can understand a physical object or an idea (especially a scientific one) mathematically, we can hardly exhaust all the possibilities and implications in the mathematical process of adding an extra dimension to our physical universe.

Philip’s description of his perception of the infinite and timeless dimension after he had crossed over is similar to a 2D person (say) trying to imagine a 3D space by supposing he or she can superimpose an infinite number of 2D universes upon and without actually touching each other. We should presume that the 2D person has no experience and cannot truly visualize a 3D space if he or she did exist. Below is a graphical illustration of a stack of 2D universes upon each other. The galaxies or stars therein are supposed to be 2D objects.

I hope the above could help enlighten that our verbal or literal expressions of ideas or mental images are often limited by our own experience and perception. However, such efforts to express ourselves is essential in our exchange and communication. Hopefully, we can then stimulate one another for a better understanding of both the external world(s) and our internal consciousness.

Math as a Gateway of Understanding

This article intends to illustrate how mathematics can become a gateway of understanding. For simplicity, I would concentrate on the idea of distance and the effects of adding or removing the relevant physical dimensions on the understanding of it.

Let’s consider the Distanceformula from 1 to 4 dimensions as follows:

Since human beings are basically living in a 3D world, the physical distance (d) between 2 points (x₁, y₁, z₁) and (x₂, y₂, z₂) is usually given by

If we try to extend the foregoing formula by adding 1 more dimension to make it 4D, then we may arrive at a similar distance formula as follows:

4D formula

The difficulty here is how can we understand the additional dimension? As 3D persons, it’s difficult to think of any realistic 4^th dimension physically as we could NOT think of a dimensional axis (W) in addition to the existing X, Y and Z axes. As an illustration, it is relatively easier to downgrade a 3D to a 2D situation. We simply drop the Z axis to form the X-Y plane. Accordingly, the distance simply becomes

Similarly, we can downgrade a 2D to 1D and got

Most important, we don’t have a problem of visualizing the said distances as straight lines in their respective dimensions (i.e. either 2D or 1D). However, we could only somehow presume that in a 4D situation the distance is still a STRAIGHT LINE. Yet, how such a 4D straight line exists would be an enigma as it CANNOT be another 3D straight line in our world!

We often heard that we are living in a 4D world comprising space and time where time is usually referred to as the 4^th dimension. Yet, can we simply replace w by t in the above 4D formula? The answer is “NO” because it doesn’t make any realistic sense in the physical world.

Indeed, time is an element of speed which is the rate of change of distance over time. As such, we don’t usually think of time as another “solid” dimension. As this article is aimed at illustrating how mathematics can become a gateway of understanding, it suffices to note that one can move forward, backward, etc… freely in space. In contrast, time flows in only one direction (the arrow of time) always pointing towards the future. Besides, relativity theories predict that gravity and speed cause time dilation and length contraction. Besides, time cannot become a standard spatial axis. Hence, time could be considered as imaginary or subjective.

Points to ponder

Can we formulate a distance formula without considering its dimensions? The answer is NO as illustrated above. Strictly speaking, it’s a complicated matter when one try to understand distance in the various dimensions. For example, a 3D space is different from a 2D space. 2D physical objects do not exist in our universe because all real matter requires a minimum of 3 spatial dimensions and volume to occupy space. 1D or 2D concepts are in fact mathematical concepts to help build mental models or simplifications to understand and calculate complex 3D reality by ignoring the depth dimension (say).

When we are watching a shadow play, the figures projected on the screen could move up and down or left or right on the screen. The movements are controlled by the puppeteers behind the screen. Hence, the defining movements happen behind the screen in a 3D space.

Even though we may imagine that some kind of 2D persons did exist in the screen’s 2D plane, we have to admit that they cannot move as freely and intentionally as watched in shadow play. For instance, if these 2D persons are presumed to be “solid” in their own right, person A on the screen have to move around person B on the screen without detaching itself from the screen (as the puppeteers do otherwise) in order to overcome B, say. Similarly, if there is a solid ground on the screen, a group of persons thereon could only move in a lineup order. Any one in the said line could only “jump” or “fly” over the others in order to change position supposing that there is an atmosphere above the ground.

I hope you can grasp the complexity of the movement as discussed in the previous paragraph in the various dimensions when we try to add or remove the dimensions of space in question. Let me give one more example here. Let’s imagine that we are 1D entities with a similar rationality as who we truly are. How could we make an extrapolation to a 2D world? As 1D entities, we must assume that we have no experience or perception of a physical 2D world. We can also suppose that our 1D world is solid and sufficient for our activities and there exist NO transcendental capability for us to transcend our bodies to a different world (or universe). Thus, we could only imagine mentally that we can add another dimension to our existing dimension. The question is how? One way to do it is by supposing that there are an infinite numbers of 1D (linear) worlds (universes) besides our particular one. The  figure is produced by AI Gemini

Also, these 1D worlds could never touch or connect with each other (otherwise we can easily move into a 2D world consequently). However, they can be packed so closely together to form a 2D plane in our imagination which can only be a vague and loosely defined concept by us as 1D entities.

Coming back to our 3D reality, we could imagine more or less similarly when we try to expand our 3D to a 4D world while time is unchanged. Then, one can either postulate some kind of mathematical concept to formulate the 4D ideas (distance, say) or make up funny but unreal ideas based on our 3D experience and perception.

While we can understand a physical object or an idea (especially a scientific one) through the gateway of mathematics, we can hardly exhaust all the possibilities and implications in the mathematical process of adding or removing dimensions in our investigation and pondering. Obviously, one can understand our world from various disciplines, faculties or reasonable spectacles available to us. I believe that such avenues for understanding are valuable and meaningful in their respective field and vision. Perhaps, each one of us would gain a beneficial understanding thereby. However, it would be difficult if not impossible for us to have a complete and all-compassing understanding of a subject in our individual capacity.

Takeaway

1. Math is a useful tool aiding our understanding of physical phenomena and concepts.

2.  Math does has its own limitation when applied typologically to the formation of ideas or notions.

3. Our understanding would become more realistic and reliable when we know the limits of our tool, e.g. math.

4. Our understanding can be enhanced when we can delineate the context of our investigation into the physical, philosophical and spiritual dimensions.

5.  The uses of our understanding should also be properly delineated.

The meaning of church history, scripture, doctrine and liturgy

In Christian theology, church history, scripture, doctrine, and liturgy form the four foundational pillars that shape the faith, practice, and identity of the Christian church. Together, they explain where the faith came from, what it believes, and how it is practiced.

Scripture (The Foundation)

•  Definition: The collection of canonical books (Old and New Testaments) written by human authors under the inspiration of the Holy Spirit.

•  Authority: Serves as the ultimate norm and standard for Christian faith, morality, and practice.

•    Function: Relates the history of salvation, from creation to the life of Jesus Christ and the early Church.

Doctrine (The Teaching)

•  Definition: The formal interpretation and systematization of truths found in Scripture.

•  Formulation: Developed over centuries through church councils, creeds (like the Nicene Creed), and theological writings to defend against heresies.

•  Function: Provides clarity, preserves orthodoxy, and teaches believers the core tenets of the faith (e.g., the Trinity, the Resurrection).

Liturgy (The Worship)

•   Definition: Derived from the Greek word leitourgia (meaning "public work" or "service"), it refers to the ritual framework of public worship.

•    Elements: Includes sacraments (like Baptism and Communion), prayers, Bible readings, and seasonal calendars (like Advent and Lent).

•    Function: Shapes the spiritual life of believers by transforming abstract doctrines into a living, communal, and sensory experience.

Church History (The Context)

•   Definition: The chronological story of the Christian Church from its New Testament origins to the modern era.

•  Scope: Covers the spread of the gospel, theological controversies, institutional splits (like the Great Schism and the Reformation), and global missionary movements.

•  Function: Helps believers understand how current practices, denominations, and cultures were shaped by the faithful actions and mistakes of past generations.

Indeed, the 4 pillars are also 4 gateways through which one may understand and commit oneself to Christianity. Take history as an example, when we look at how Jesus’ crucifixion and resurrection have established the foundation of Christianity and how the followers, the apostles in particular, devoted themselves to the preaching of the gospel, we would naturally ask ourselves, “Do I truly believe in Jesus’ resurrection and the promise of eternal Life?” and “How can I partake in the spreading of the gospel?”, etc… These are simple and yet far-reaching questions. Perhaps, each of us would get an individual answer/quest which is personal and particular according to the will of our God and Savior. May the Lord enlighten and help us. Amen.

Howard

教會歷史，經典，信條及禮儀的具體意義

在基督教的傳統中，教會歷史、經典、信條及禮儀共同構成了信仰的四大支柱。它們不僅是宗教實踐的規範，更是連結過去、現在與未來的信仰通道。

我網搜了相關主題，得到以下為這四個核心要素的具體意義與核心價值的分析：

一、 教會歷史（Church History）的具體意義

教會歷史大致可指從耶穌基督復活升天、聖靈降臨（使徒時代）開始，直到今日全球教會的發展歷程。

·        信仰身份的根源： 歷史讓信徒明白自己的信仰從何而來。它證明了基督信仰不是憑空虛構的，而是承襲了兩千年的集體經歷的屬靈產業。

·        見證信仰的生命力： 歷史記錄了教會在面對羅馬帝國逼迫、政治動盪和文化變遷時，如何藉聖靈的引導和信徒的委身得以承傳，展現信仰的生命力。

·        歷史的借鑒與啟導： 教會歷史並不完美，其中包含了宗教戰爭、權力腐化、分裂等負面歷史。研究歷史能幫助當代教會引以為戒，避免重蹈覆轍，並學習先賢在困境中的智慧和委身。

二、 經典（Scripture, The Bible）的具體意義

教會核心經典是聖經，是經過早期教會「正典化」（Canonization）過程所確立為 神的話語和啟示。

·        信仰與生活的最高權威： 經典是檢驗一切神學思想、道德行為和教會治理的最高標準（參考Sola Scriptura，唯獨聖經）。

·        神的自我啟示： 聖經記載了 神與人類立約和聯繫的歷史，尤其是透過耶穌基督成就的救贖計劃。它是信徒認識 神心意及與 神聯繫的核心媒介。

·        群體塑造的基石： 經典透過文字塑造了基督徒的世界觀、價值觀與生活指引，使全球不同文化的信徒擁有共同的價值體系。

三、 信條（Creeds and Confessions）的具體意義

信條（或稱信經、信仰宣言，如《使徒信經》、《尼西亞信經》）是教會為了回應歷史挑戰，將聖經中的核心真理，進行高度濃縮與系統化的摘要。

·        辨別真偽、抵擋異端： 在早期教會，許多錯誤教導（異端）興起。信條的出現是為了劃清真理的界線，保護信徒不被錯誤的觀念誤導。

·        信仰認同的宣告： 當群體一同認同「信條」時，是在向世界、向彼此宣告「這就是我們共同持守的信仰核心」。它是教會合一的重要指引。

·        神學教育與信仰標準： 歷史上，信條常被用作初信者接受洗禮前的基要真理教材，確保其信仰建立在正統的根基上。

四、 禮儀（Liturgy）的具體意義

禮儀是指教會在崇拜中所使用的規定程序、儀式、象徵物以至是節期（如聖餐禮、洗禮、教會年曆等）的實踐。

·        抽象真理的具體化： 禮儀將深奧的概念轉化為「看得到、摸得到、聞得到、聽得到」的感官經歷。例如，聖餐禮透過餅與杯，讓信徒親身體會基督身體的擘開與寶血的流出。

·        塑造屬靈的習慣： 禮儀具有重複性。透過不斷參與神聖的儀式，真理和教義會刻進信徒的內心記憶中，從而塑造個人的生命品格與生活規律。

·        參與 神的作為： 透過禮儀（如復活節、聖誕節等教會年曆的循環），信徒每年都在重新經歷耶穌基督的降生、受死、復活與升天等，具體地引導信徒融入 神的作為之中。

總結

這四者相輔相成，並互相關聯，是不可分割的，因為經典提供了真理的基礎；信條將真理整合為清晰的教義；禮儀將信念化為日常的實踐與崇拜的指引；而教會歷史則是這一切在時空長河中不斷流傳、發展和整合的維度(dimension)。從哲學的角度，以上四者皆可視為進入信仰本體的四度門戶，信徒大概都要從中進入，才能掌握和經驗信仰的真確性。它們也是世人尋求認識真道和真神的有效途徑，配以默想辨識，以至是個人的靈性經驗，好能認識真理，並與真神建立緊密的關係，謹供參考，願 主光照和幫助大家！

Howard